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- 1. Government Property NOT FOR SALE NOT Mathematics Quarter 2 - Module 3 Solving Problems Involving Polynomial Functions Department of Education ● Republic of the Philippines 10
- 2. Mathematics- Grade 10 Alternative Delivery Mode Quarter 2 – Module 3: Solving Problems Involving Polynomial Functions First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalty. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education DepEd Secretary: Leonor M. Briones Development Team of the Module Author/s: Stephen Edd T. Navarro, MACDDS Reviewers: Rhodel A. Lamban, PhD Elbert R. Francisco, PhD Eleonor A. Villamor, EdD Illustrator and Layout Artist: Stephen Edd T. Navarro, MACDDS Management Team Chairperson: Arturo B. Bayocot, PhD, CESO III Regional Director Co-Chairpersons: Victor G. De Gracia Jr., PhD, CESO V Asst. Regional Director Randolph B. Tortola, PhD, CESO IV Schools Division Superintendent Shambaeh A. Usman, PhD Assistant Schools Division Superintendent Mala Epra B. Magnaong, Chief - CLMD Neil A. Improgo, PhD, EPS - LRMS Bienvenido U. Tagolimot Jr., PhD, EPS - ADM Members: Elbert R. Francisco, PhD, Chief - CID Rhodel A. Lamban, PhD, EPS - Mathematics Rejynne Mary L. Ruiz, PhD, LRMDS Manager Jeny B. Timbal, PDO II Shella O. Bolasco, Division Librarian II Printed in the Philippines by Department of Education – Division of Bukidnon Office Address: Fortich St., Sumpong, Malaybalay City Telephone: (088) 813-3634 E-mail Address: bukidnon@deped.gov.ph
- 3. Mathematics Quarter 2 - Module 20 SOLVING PROBLEMS INVOLVING POLYNOMIAL FUNCTIONS This instructional material was collaboratively developed and reviewed by educators from public school. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at bukidnon@deped.gov.ph. We value your feedback and recommendations. Department of Education-Division of Bukidnon ● Republic of the Philippines
- 4. Table of Contents PAGE What This Module is About How to Learn from this Module Icons of this Module What I Need to Know What I Know Lesson 1: Solving Problems Involving Polynomial Functions Through Evaluation What I Need to Know..................................................................................................... 1 What I Know..................................................................................................................... 1 What’s In............................................................................................................................ 3 What’s New ................................................................................................................... 3 What Is It ........................................................................................................................... 5 What’s More .................................................................................................................... 6 What I Have Learned..................................................................................................... 8 What I Can Do................................................................................................................. 9 Assessment...................................................................................................................... 9 Additional Activities........................................................................................................ 11 Lesson 2: Solving Problems Involving Polynomial Functions Through Factoring What I Need to Know..................................................................................................... 13 What I Know..................................................................................................................... 13 What’s In............................................................................................................................ 15 What’s New ................................................................................................................... 15 What Is It ........................................................................................................................... 17 What’s More .................................................................................................................... 19 What I Have Learned..................................................................................................... 20 What I Can Do................................................................................................................. 21 Assessment...................................................................................................................... 22 Additional Activities........................................................................................................ 24 Lesson 3: Modeling Polynomials (Day 3 and 4) What I Need to Know..................................................................................................... 26 What I Know..................................................................................................................... 26 What’s In............................................................................................................................ 28 What’s New ................................................................................................................... 28 What Is It ........................................................................................................................... 29 What’s More .................................................................................................................... 33 What I Have Learned..................................................................................................... 34 What I Can Do................................................................................................................. 35 Assessment...................................................................................................................... 36
- 5. Additional Activities........................................................................................................ 37 Summary .................................................................................................................................................. 39 Assessment: (Post-Test) ...................................................................................................................... 40 Answer Key................................................................................................................................................ 42 References 46 .......................................................................................................................................................................
- 6. What This Module is About Welcome to Mathematics 10. This is the Alternative Delivery Mode Module (ADM) in solving problems involving polynomial functions. On this module, you will encounter different kinds of real-life problems that’s involves polynomial functions and ways on how to solve them. This module will also test your previous knowledge about polynomials. This module was collaboratively designed, developed and reviewed by educators both from public and private institutions to assist you, the teacher or facilitator in helping the learners meet the standards set by the K to 12 Curriculum while overcoming their personal, social, and economic constraints in schooling. This learning resource hopes to engage the learners into guided and independent learning activities at their own pace and time. Furthermore, this also aims to help learners acquire the needed 21st century skills while taking into consideration their needs and circumstances. For the Facilitators: As a facilitator you are expected to orient the learners on how to use this module. You also need to keep track of the learners' progress while allowing them to manage their own learning. Furthermore, you are expected to encourage and assist the learners as they do the tasks included in the module. For the Learners and Parents: This module was designed to provide you with fun and meaningful opportunities for guided and independent learning at your own pace and time. You will be enabled to process the contents of the learning resource while being an active learner. If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Always bear in mind that you are not alone. We hope that through this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. You can do it!
- 7. How to Learn from this Module To achieve the desired objectives, you are to do the following: • Take your time reading the lessons carefully. • Follow the directions and/or instructions in the activities and exercises diligently. • Answer all the given tests and exercises. Icons of this Module What I Need to Know This part contains learning objectives that are set for you to learn as you go along the Module each day/lesson. What I know This is a pre-test assessment as to your level of knowledge to the subject matter at hand, meant specifically to gauge prior related knowledge. What’s In This part connects previous lesson with that of the current one. What’s New An introduction of the new lesson through various activities, before it will be presented to you. What is It These are discussions of the activities as a way to deepen your discovery and under- standing of the concept. What’s More These are follow-up activities that are in- tended for you to practice further in order to master the competencies. What I Have Learned Activities designed to process what you have learned from the lesson. What I can do These are tasks that are designed to show- case your skills and knowledge gained, and applied into real-life concerns and situations. Assessment This is a task which aims to evaluate your level of mastery in achieving the learning competency. Additional Activities In this portion, another activity will be given to you to enrich your knowledge or skill of the lesson learned. This also tends retention of learned concepts. Answer Key This contains answers to all activities in the module.
- 8. What I Need to Know This module was designed and created especially for you. This will help you understand polynomial functions deeper and solve real-life problems involving polynomial functions. The scope of this module permits it to be used in many different learning situations. The language used recognizes the diverse vocabulary level of students. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using. After going through this module, you are expected to: 1. identify the necessary information to be used in solving real-life problems involving polynomial functions; 2. solve real-life problems that can be modeled with polynomial functions through evaluation; and 3. apply polynomial functions to real-life problems and produce a concrete object that satisfies the condition given on the problem.
- 9. What I Know Let us find out how much you already know about the topics in this module. Answer all items. Choose the letter of the correct answer to each question. Write the chosen letter on a separate sheet of paper. 1. A furniture dealer determines that its profit, P, in thousands of pesos, can be modeled by the function P(x) = 0.0012x3 + 2x – 25, where x represents the number of furniture sold. What is the profit when x = 50? A. Php 200, 000 B. Php 225, 000 C. Php 250, 000 D. Php 275, 000 2. In this school year, 50 high school students participated in one-week Human vs Zombie game. The number of zombies, Z, after d days of playing can be modeled by the function Z(d) = 5d3 – 3d2 + 15. How many days did it take for the number of zombies to reach 565? A. 4 B. 5 C.6 D. 7 3. Faye is going to throw a rock from the top of a trail overlooking the ocean. When she throws the rock upward from 160 ft above the ocean, the function h(t) = -16t2 + 48t + 160 models the height, h, of the rock above the ocean as a function of time, t. Find the height of the rock at t = 1.5 seconds. A. 148 ft B. 164 ft C. 180 ft D. 196 ft 4. The estimated number of electric vehicles, V (in thousands), in the Philippines can be modeled by the polynomial function: V(y) = 0.15128y3 – 3.28234y2 + 23.7565y – 2.041 where y represents the number of years after the year 2005. Find the estimated number of electric vehicles at the end of the year 2007. Round up to the nearest thousands. A. 32, 000 B. 33, 000 C. 34, 000 D. 35, 000 5. The weight, w (in kilograms), of a certain fish can be modeled by w = 0.00304x3, where x is the length of the fish in inches. What is the weight of a 12-inch fish? A. 5.00 kg B. 5.25 kg C. 5.50 kg D. 5.75 kg 6. The profit, P (in millions of pesos), for a flash drive manufacturer can be modeled by P = n3 + 6n2 - 7n where n (in millions) is the number of flash drive produced. How many flash drives was produced if the profit reached 60 million pesos? A. 3, 000, 000 B. 4, 000, 000 C. 5, 000, 000 D. 6, 000, 000 7. The volume, V (in cubic feet), of a rectangular hydraulic block is modeled by the polynomial function V(w) = 6w3 – 19w2 – 52w where w is the width (in feet) of the hydraulic block. What is the width of the hydraulic block if its volume is 15 cubic feet? A. 4 feet B. 5 feet C. 6 feet D. 7 feet 8. During a 20-year period, the amount (in millions of pesos) of school supplies (S) sold can be modeled by S(t) = - 20t3 + 25t2 – 280t + 3320, where t is in years. Find the amount of school supplies sold in 5 years. A. Php 35, 000, 000 B. Php 40, 000, 000 C. Php 45, 000, 000 D. Php 50, 000, 000
- 10. For Item number 9 – 11, refer to the problem below. On a glass factory, paperweights are created by pouring molten glass into molds. Each mold is a rectangular prism with a height 3 cm greater than the length of each side of its square base. Each mold holds 112 cubic centimeters of glass. 9. If x represents the length of each side of the square base, which of the following mathematical model satisfy the condition of the above problem? A. (x + 3) (x + 3) (x) = 112 B. (x + 3) (x) (x) = 112 C. (x + 3) (x) = 112 D. 112 (x + 3) = (x) (x) 10. What is the height of the glass mold? A. 4 cm B. 5cm C. 6 cm D. 7 cm 11. What are the dimensions of the glass molds? (Follow V = LWH) A. 4 cm by 5 cm by 8 cm B. 4 cm by 4 cm by 7 cm C. 5 cm by 5 cm by 8 cm D. 5 cm by 8 cm by 8 cm For Item number 12 – 15, refer to the problem below. A construction company was hired to build a swimming pool for public use. The client wants the depth of the pool to be 2 meters less than the width and the length is 10 meters more than the width. The client also specified that the pool must have a water capacity of 595 cubic meters. 12. If w represents the measurement of the width, which of the following mathematical model satisfy the condition of the above problem? A. (w - 2) (w + 2) (w) + 10 = 595 B. (w - 2) (w - 10) (w) = 595 C. (w - 2) (w + 10) (w) = 595 D. (w - 10) (w + 2) (w) = (595 13. How deep is the pool? A. 3 meters B. 4 meters C. 5 meters D. 6 meters 14. How long is the pool? A. 15 meters B. 17 meters C. 19 meters D. 21 meters 15. What are the dimensions of the swimming pool? (Follow V = LWH) A. 17 m by 5 m by 5 m B. 12 m by 2 m by 7 m C. 12 m by 5 m by 8 m D. 17 m by 7 m by 5 m
- 11. 1 Lesson 1 Solving Problems Involving Polynomial Functions Through Evaluation (Day 1) What I Need to Know After going through this lesson, you are expected to: 1. identify the necessary information to be used in solving real-life problems involving polynomial functions; and 2. solve real-life problems that can be modeled with polynomial functions through evaluation. What I Know Let us find out how much you already know about the topic on this lesson. Answer all items. Choose the letter of the correct answer to each question. Write the chosen letter on a separate sheet of paper. For Items number 1 – 5, Refer to the problem below. The estimated number of electric vehicles, V (in thousands), in the Philippines can be modeled by the polynomial function: V(y) = 0.15128y3 – 3.28234y2 + 23.7565y – 2.041 where y represents the number of years after the year 2005. 1. What is the value of y at the end of year 2008? A. 3 B. 4 C. 5 D. 6 2. Based on the problem, V represents __________. A. the estimated number of electric vehicles. B. the number of years after the year 2005. C. the estimated number of the modeled polynomial function. D. the number of vehicles present in the Philippines. 3. At what year after 2005 will the estimated number of electric vehicles be calculated if y = 5? A. 2005 B. 2007 C. 2008 D. 2010 4. What is V(y) if the value of y = 2? (Round off to the nearest thousand). A. 33, 000 B. 34, 000 C. 35, 000 D. 36, 000
- 12. 2 5. What is the estimated number of electric vehicles at the end of year 2008? (Round off to the nearest thousand). A. 45, 000 B. 46, 000 C. 47, 000 D. 48, 000 For Items number 6 – 10, Refer to the problem below. The volume, V (in cubic feet), of a rectangular hydraulic block is modeled by the polynomial function V(w) = 2w3 + 11w2 – 12w where w is the width (in feet) of the hydraulic block. 6. Based on the problem, w represents __________. A. the volume of the rectangular hydraulic block. B. the length of the rectangular hydraulic block. C. the width of the rectangular hydraulic block. D. the height of the rectangular hydraulic block. 7. The polynomial function represents ____________. A. the volume of the rectangular hydraulic block. B. the length of the rectangular hydraulic block. C. the width of the rectangular hydraulic block. D. the height of the rectangular hydraulic block. 8. What is V(w) if the value of w = 4? A. 48 ft3 B. 128 ft3 C. 176 ft3 D. 256 ft3 9. What is the volume of a 2 feet wide hydraulic block? A. 36 ft3 B. 40 ft3 C. 44 ft3 D. 48 ft3 10. The polynomial function of the rectangular block represents what equation? A. B. C. D. For Items number 11 – 15, Refer to the problem below. The profit, P (in millions of pesos), for a flash drive manufacturer can be modeled by P = 6n3 + 72n where n (in millions) is the number of flash drive produced. What would be the profit of the manufacturer if it produced 3 million flash drives? 11. Based on the problem, P represents __________. A. the number of flash drive produced. B. the profit of the flash drive manufacturer. C. the 3 million flash drives produced. D. the number of manufacturers who produce flash drives. 12. Based on the problem, if the value of n = 5 then the manufacturer produced how many flash drives? A. 5 000 B. 50 000 C. 500 000 D. 5 000 000 13. What is the value of n when the manufacturer produced 3 000 000 flash drives? A. 3 B. 300 C. 30 000 D. 3 000 000 14. What is P if the value of n = 3? A. 162 million B. 216 million C. 342 million D. 378 million 15. What would be the profit of the manufacturer if it produced 2 million flash drives? A. 54 million B. 144 million C. 192 million D. 216 million
- 13. 3 What’s In Determine the leading term, end behaviors, y –intercept, and number of turning points for each given polynomial function. 1. f(x) = (x2 – 3)2 (x2 – 1)2 2. f(x) = (x2 – 7) (x – 1)2 (x – 2)3 3. f(x) = -x3 – x2 + 2x What’s New Let’s Explore 1. Suppose that the Town of Talakag discovered wild boars near the town’s outskirts in 2010. So, they began tracking the number of wild boars near the town outskirts each year and the following chart shows how many wild boars, B, were present each year, x, after 2010. x (years) 1 2 3 4 5 5.5 6 B (number of wild boars) 1 4 10 20 35 45 56 Some Zoologist were called in to analyze the population trends of the wild boars in hopes to keep the population under control, and they found the data in the chart can be modeled using the function: where B is the number of wild boars on x years. Were you able to apply all the necessary concepts and properties in graphing each function? The next activity will let you see the connections of these mathematics concepts to real – life situations.
- 14. 4 Questions/ Tasks: a. Using the graph, what is the approximate number of wild boars after 3.5 years? b. Compare the number of wild boars in year 5.5 as given on the table to the number of wild boars given using the polynomial functions. Why are these values not exactly the same? c. Do you think you can use the polynomial function to predict the number of wild boars on the 10th year? Why or why not? Let’s try this Answer the following: 1. The number of cable TV systems sold after year 2010 can be modeled by the polynomial function C(t) = -43.2t2 + 1343t + 790, where t represents the number of years since 2010. a. How many cable TV systems sold after the end of 2015? b. After 10 years from 2010, what is the expected number of cable TV systems sold? 2. Suppose that the average fuel (in Liters) consumed by individual vehicles monthly in Philippines is modeled by the cubic equation F(t) = 0.025t3 – 1.5t2 + 18.25t + 148, where t in the number of months. a. What is the average fuel consumed by individual vehicles after half a year? b. After 1 year, what is the expected average fuel consumed by individual vehicles? Were you surprised that polynomial functions can be applied in real – life problems and also have practical uses? What do you need to solve these problems? Enjoy learning as you proceed to the next section.
- 15. 5 3. An owner of a certain resort decided to remove the pollutants on their swimming pool. The cost C (in pesos) of removing p percent of pollutants from the swimming pool is given by the polynomial function C(p) = 25x2 – 50x + 24500. a. What is the cost of removing 20% of the pollutants from the swimming pool? b. If the owner decided to remove all the pollutants from the swimming pool, how much would it cost? What Is It Your previous lesson has defined Polynomial Function as a function of the form P(x) = an xn + an-1 xn-1 + … + a1 x + a0 where an are real numbers called coefficients and n is a positive integer, and an xn is the leading term, an is the leading coefficient, a0 is the constant term and an ≠ 0. Since the problems involves polynomial functions then we could use many ways on solving these kinds of problems. Here we are going to discuss and solve problems together step by step. The problems presented below are things that we could relate to real-life situations. Example 1: Suppose a windmill generates power based on the speed of the wind. This is represented by the polynomial function , where s represents the speed of the wind in kilometers per hour. If the wind speed is 15 kilometers per hour, how many units of power, P(s), can the windmill generate? Procedure: Step 1 , s = 15kph Rewrite the given. Step 2 Find P (15) Identify what is asked on the problem. We have P (15) since it is asking on how many units of power can the wind mill generate if the windspeed is 15 kph. Step 3 , Replace all the variable s with 15, since s = 15 based on the given. Step 4 Simplify the equation. Remember that (15)4 = (15((15)(15)(15) = 50625. Step 5 P(15) = 50.625 By dividing 50625 by 1000, we arrive on the result 50.625. Answer: The windmill can generate 50.625 units of power if the wind speed is 15 kilometers per hour.
- 16. 6 Example 2: The number of citizen (N) of Barangay San Isidro in t years is predicted to increase and can modeled by the function N(t) = 7t4 – 4t3 + 150t + 17500. After 3 years, how many citizens will be residing now at Barangay San Isidro? Procedure: Step 1 N(t) = 7t4 – 4t3 + 150t + 17500, t = 3 years Rewrite the given. Step 2 Find N(3) Identify what is asked on the problem. We have N(3) since it is asking on how many citizens will be residing at Barangay San Isidro after 3 years. Step 3 N(3) = 7(3)4 – 4(3)3 + 150(3) + 17500 Replace all the variable t with 3, since t = 3 based on the given. Step 4 N(3) = 7(81) – 4(27) + 150(3) + 17500 Simplify the equation. Remember that (3)4 = (3)(3)(3)(3) = 81 and (3)3 = (3)(3)(3) = 27. This is to make multiplication easier. Step 5 N(3) = 567 – 108 + 450 + 17500 Simplify each term. This is by multiplying 7 by 81 which is equal to 567, 4(27) = 108, and 150(3) = 450. Step 6 N(3) = 18, 409 Add all values to get the final result. Thus, 567 – 108 + 450 + 17 500 is equal to 18 409. Answer: There will be 18, 409 citizens residing at Barangay San Isidro after 3 years. What’s More Activity 1: Ref! Substitute! Direction: Answer the following question through evaluating each given polynomial function. (3 points each) 1. The polynomial function, w = 0.0071d3 – 0.09d2 + 0.48d, models the weight of the ideal round-cut diamond where w is the diamond’s weight (in carats) and d is the diameter (in millimeters). Based on the model, find the weight of the diamond with a diameter of 20 millimeters. 2. A drugstore that sells a certain brand of Vitamin C tablets predicts that their profit P can be modeled by the polynomial function P(a) = -50a3 + 2400a2 –
- 17. 7 2000, where a is the amount spent on advertising (in thousands of pesos). What is its profit if a = 16? 3. Suppose that the total number attendance A (in thousands) at NCAA women’s basketball games is modeled by the polynomial function A = -1.95t3 + 70.1t2 – 188t + 2150, from 2000 to 2018, where t is the number of years since 2000. Find the attendance for the year 2009. 4. The number of eggs, E(x), in a female butterfly is a function of her abdominal width, x, in millimeters, modeled by E(x) = 14x3 – 17x2 – 16x + 34. How many eggs can the butterfly carry if the width of her abdomen is 4 millimeters? 5. A pyramid can be formed using same sized balls. For example, 3 balls can be arranged in a triangle, then the fourth ball placed in the middle on top of them. The total number of balls used can be modeled by the polynomial function, , where n is the number of balls on each side of the bottom layer. What is the total number of balls used to form a pyramid if there are 5 balls on each side of the bottom layer? Activity 2: Replace Me! Direction: Answer the following question through evaluating each given polynomial function. (3 points each) 1. The profit P (in millions) of a certain brand of energy drink in a year is represented by the polynomial equation P(t) = 5t3 + 20t2 – 30t – 240 where t is the number of years. How much is the profit after 6 years? 2. If you look at the cross sections of a honeycomb, you will see patterns of hexagons. Suppose that the number of hexagon H inside a honeycomb is given by the polynomial function H(y) = 3y2 – y + 1 where y is the number of layers inside the honeycomb. If there are 5 layers inside the honeycomb, how many hexagons are present inside? 3. In 2010, the population P of a certain municipality is modeled by the function P(t) = 6t3 – 700t2 + 5000t + 140000 where t is the number of years after 2010. What is the population of the municipality after 2019? 4. A windmill generates power based on the speed of the wind. This is represented by the function , where s represents the speed of the wind in kilometers per hour. How many units of power, P(s), can the windmill generate if the wind speed is 10 kilometers per hour? 5. The number of cable TV systems sold after year 2010 can be modeled by the polynomial function C(t) = -43.2t2 + 1343t + 790, where t represents the number of years since 2010. How many Cable TV systems is sold at the end of 2016?
- 18. 8 What I Have Learned Direction. Chose a word or phrase from the box below to fill in the missing parts. 1. In solving problems involving polynomial function, I must first ______ the problem before listing down the given. 2. One of the effective ways to not forget the given is to _____ it down first. 3. Before solving the problem, I must identify first __________. 4. To evaluate a polynomial function, I would just simply ______ the given value to its corresponding variables then solve. 5. After knowing what is asked, I must create a __________ before I can start solving the problem. Direction. Answer the following problems. (2 points each). An owner of a certain resort decided to remove the pollutants on their swimming pool. The cost C (in pesos) of removing p percent of pollutants from the swimming pool is given by the polynomial function C(p) = 15x2 – 25x + 1500. 6. How much would it cost if you remove 20% of the pollutants in the swimming pool? 7. What is the cost of removing 30% of the pollutants from the swimming pool? 8. If the owner decided to remove half the pollutants from the swimming pool, how much would it cost? 9. How much does it cost if all the pollutants are removed? 10. If you are the owner, what will you do? Remove the pollutants by percent? Or remove all the pollutants? Why? mathematical model read write what is asked positive substitute
- 19. 9 What I Can Do Directions: Read and analyze the situation below. Then, answer the following questions. You are the CEO of a big company that sells the latest versions of gaming laptops in the whole world. You have created a polynomial function, G(x) = - 0.015x4 + 0.735x3 – 12.7x2 + 60.5x + 350, that tells you the relationship between the number of gaming laptops sold (in millions) and the time x years since 2030 (x=0 in 2030). Perform the following: a. Find the number of gaming laptop sold in year 2040. b. On every gaming laptop sold, you donate 10 pesos on a certain charity that helps those students who cannot go to school because of financial problem. How much would you donate on the charity on the year 2035? c. On solving the previous problem above, what did you do? Explain what you have done. Assessment Let us find out how much you have learned about the topic on this lesson. Answer all items. Choose the letter of the correct answer to each question. Write the chosen letter on a separate sheet of paper. For Items number 1 – 5, Refer to the problem below. The estimated number of motorcycles, V (in thousands), in the Philippines can be modeled by the polynomial function: V(y) = 24y3 – 15y2 + 15y – 440 where y represents the number of years after the year 2000. 1. What is the value of y at the end of year 2007? A. 5 B. 6 C. 7 D. 8 2. Based on the problem, V represents __________. A. the estimated number of motorcycles.
- 20. 10 B. the number of years after the year 2000. C. the estimated number of the modeled polynomial function. D. the number of vehicles present in the Philippines. 3. At what year after 2000 will the estimated number of motorcycles be calculated if y = 4? A. 2004 B. 2005 C. 2006 D. 2007 4. What is V(y) if the value of y = 4? A. 916, 000 B. 945, 000 C. 1, 325, 000 D. 1, 536, 000 5. What is the estimated number of motorcycles at the end of year 2006? A. 3, 147, 000 B. 4, 294, 000 C. 5, 481, 000 D. 6, 162, 000 For Items number 6 – 10, refer to the problem below. The volume, V (in cubic feet), of a rectangular ice block is modeled by the polynomial function V(w) = 4w3 + 13w2 – 5w where w is the width (in feet) of the ice block. 6. Based on the problem, w represents __________. A. the volume of the rectangular ice block. B. the length of the rectangular ice block. C. the width of the rectangular ice block. D. the height of the rectangular ice block. 7. The polynomial function represents ____________. A. the volume of the rectangular ice block. B. the length of the rectangular ice block. C. the width of the rectangular ice block. D. the height of the rectangular ice block. 8. What is V(w) if the value of w = 4? A. 256 ft3 B. 323 ft3 C. 434 ft3 D. 516 ft3 9. What is the volume of a 2 feet wide ice block? A. 62 ft3 B. 66 ft3 C. 70 ft3 D. 74 ft3 10. The polynomial function of the rectangular block represents what equation? A. B. C. D. For Items number 11 – 15, refer to the problem below. The profit, P (in millions of pesos), for a laptop manufacturer can be modeled by P = 3n3 + 64n where n (in millions) is the number of laptops produced. What would be the profit of the manufacturer if it produced 3 million laptops? 11. Based on the problem, P represents __________. A. the number of laptops produced. B. the profit of the laptop manufacturer. C. the 3 million laptops produced. D. the number of manufacturers who produce laptops. 12. Based on the problem, if the value of n = 5 then the manufacturer produced how many laptops? A. 6 950 B. 69 500 C. 6 950 000 D. 695 000 000
- 21. 11 13. What is the value of n when the manufacturer produced 3 000 000 flash drives? A. 3 B. 300 C. 30 000 D. 3 000 000 14. What is P if the value of n = 3? A. 192 million B. 273 million D. 342 million D. 378 million 15. What would be the profit of the manufacturer if it produced 2 million flash drives? A. 152 million B. 168 million D. 198 million D. 216 million Additional Activities Activity 3: You Complete Me! Direction: Fill up the missing parts of the solution to the problem. Copy and answer on a separate sheet of paper. Silvana was told that the volume of a certain block is modeled by the function V(x) = 5x6 – 2x5 + 7x4 – 10x3 + 14x2 – 8x + 8, where x is the length in meters of the certain block. Find the volume of a certain block with a length of 2 meters. Procedure: Step 1 1. _____________ 2. _____________ Rewrite the given. Step 2 3. _____________ Identify what is asked on the problem. Step 3 4. V (__) = 5(__)6 – 2(__)5 + 7(__)4 – 10(__)3 + 14(__)2 – 8(__) + 8 5. ________________ since x = 2 based on the given. Step 4 6. V (2) = 5(__) – 2(__) + 7(__) – 10(__) + 14(__) – 8(__) + 8 Simplify the equation. Step 5 7. V(2) = __ - __ + __ - __ + __ - __ + __ Simplify each term. Step 6 8. V (2) = ____ Add all values to get the final result. 9 – 10. Answer: ______________________________________. Activity 4: Ref! Substitute! Me Again! Direction: Answer the following question through evaluating each given polynomial function. (3 points each) 1. A drugstore that sells a certain brand of Vitamin C tablets predicts that their profit P can be modeled by the polynomial function P(a) = -50a3 + 2400a2 – 2000, where a is the amount spent on advertising (in thousands of pesos). What is its profit if a = 10?
- 22. 12 2. The number of eggs, E(x), in a female butterfly is a function of her abdominal width, x, in millimeters, modeled by E(x) = 14x3 – 17x2 – 16x + 34. How many eggs can the butterfly carry if the width of her abdomen is 2 millimeters? 3. The profit P (in millions) of a certain brand of energy drink in a year is represented by the polynomial equation P(t) = 5t3 + 20t2 – 30t – 200 where t is the number of years. How much is the profit after 3 years? 4. If you look at the cross sections of a honeycomb, you will see patterns of hexagons. Suppose that the number of hexagon H inside a honeycomb is given by the polynomial function H(y) = 3y2 – y + 1 where y is the number of layers inside the honeycomb. If there are 7 layers inside the honeycomb, how many hexagons are present inside? 5. In 2010, the population P of a certain municipality is modeled by the function P(t) = 6t3 – 700t2 + 5000t + 14000 where t is the number of years after 2010. What is the population of the municipality after 2015?
- 23. Lesson 2 Solving Problems Involving Polynomial Functions Through Factoring (Day 2) What I Need to Know After going through this lesson, you are expected to: 1. identify the necessary information to be used in solving real-life problems involving polynomial functions; and 2. solve real-life problems that can be modeled with polynomial functions through factoring; What I Know Let us find out how much you already know about the topic on this lesson. Answer all items. Choose the letter of the correct answer to each question. Write the chosen letter on a separate sheet of paper. For Items number 1 – 5, Refer to the problem below. Mhaica is a successful agriculturist. She created hybrid crops that help a lot of people in a certain province. Due to her creation, the businesses that involves the said hybrid crop started to boom. The polynomial function P(x) = x3 + 2x2 – 33x – 5 represents the profit (in thousands of dollars) of a certain company that processes the said hybrid crop where x is the number of hybrid crops processed in thousands. Determine the number of hybrid crops processed when the company profited 85 thousand dollars. 1. What is asked on the problem? A. Find the number of hybrid crops processed. B. Determine the number of hybrid crops processed when the company profited 85 thousand dollars. C. Solve for the profit when 85 thousand hybrid crops are processed. D. Find P(x) when x is equal to 85 thousand dollars. 2. Based on the problem, P(x) represents __________. A. the number of hybrid crops processed. B. the number of hybrid crops Mhaica created. C. the profit of a company that processes hybrid crops. D. the profit Mhaica earned as the creator of the hybrid crops.
- 24. 13 3. How many hybrid crops were processed when the company profited 85 thousand dollars? A. 3 000 B. 4 000 C. 5 000 D. 6 000 4. What number should be substituted to P(x) based on the problem? A. 85 B. 850 C. 8, 500 D. 85, 000 5. What is the degree of the polynomial presented on the problem? A. 2nd B. 3rd C. 4th D. 5th For Items number 6 – 10, Refer to the problem below. A recording company has determined its daily revenue, R, by the polynomial function R(n) = 5n3 + 125n2 – 1000n, where n represents the number of records sold. If the company earns a revenue of 7, 500 pesos on a certain day, how many records were sold? 6. Based on the problem, n represents __________. A. polynomial function used to compute the revenue B. daily revenue of the recording company C. number of records sold D. revenue of a certain day which is 7, 500 pesos 7. The polynomial function represents ____________. A. polynomial function used to compute the revenue B. daily revenue of the recording company C. number of records sold D. revenue of a certain day which is 7, 500 pesos 8. If the company earns a revenue of 7, 500 pesos on a certain day, how many records were sold? A. 5 B. 10 C. 20 D. 30 9. What number should be substituted to P(x) based on the problem? A. 75 B. 750 C. 7, 500 D. 75, 000 10. Based on the problem, R(n) represents __________. A. polynomial function used to compute the weekly revenue B. daily revenue of the recording company C. number of records sold D. revenue of a certain day which is 7, 500 pesos For Items number 11 – 15, Refer to the problem below. A box is to be shipped abroad. The volume (in cubic feet) of the box can be expressed as the product of its three dimensions which can be represented by the function V(w) = w3 – 2w2 – 29w – 2, where w is the measure of its width in feet. What is the width of the box if the volume of the box is 40 ft3? 11. Based on the problem, V(w) represents __________. A. measure of its width in feet B. width of the box which is 40 ft3 C. dimensions of the box D. volume of the box in cubic feet
- 25. 14 12. If the volume of the box is 40 ft3, what is its width? A. 5 ft B. 6 ft C. 7 ft D. 8 ft 13. The polynomial function represents ____________. A. measure of its width in feet B. width of the box which is 40 ft3 C. dimensions of the box D. volume of the box in cubic feet 14. Based on the problem, w represents __________. A. measure of its width in feet B. width of the box which is 40 ft3 C. dimensions of the box D. volume of the box in cubic feet 15. What is the degree of the polynomial presented on the problem? A. 2nd B. 3rd C. 4th D. 5th What’s In Evaluate the following polynomials function with its given value. 1. f(x) = 3x2 – 5x + 7; f (3). 2. H(s) = s3 + 4s2 – 5s + 25; H (2). 3. V(r) = 6r4 – 11r2 + 150; V (4). What’s New Let’s Explore As a plant lover, you have a collection of different kinds of cactus and succulents. A friend of yours invited you to attend a seminar of #Richmindset Corporation. There, you’ve learned how to do business. Now, you decided to sell your plants and you have profited well while doing what you love which is planting. You’ve learned that your earnings (in hundreds of pesos) can be modeled by the function E(p) = 10p3 - 51p2 + 127p, where p is the number of set of plants sold. Were you able to apply all the necessary concepts in evaluating polynomial function? The next activity will let you see another way of solving problems involving polynomial functions.
- 26. 15 Questions/ Tasks: 1. How much would you earn by selling 2 sets of plants? 2. If you earned 4,200 pesos, how many sets of plants did you sold? 3. Based on the second question, what is given? The money you’ve earned by selling sets of plants or the number of sets of plants sold? 4. Still based on the second question, what did you use for substitution? 4200 or 42? Explain your answer. Let’s try this Read the following problems and answer the following questions: 1. The weight, w (in pounds), of a certain fish can be modeled by w = 2x2 – 5x – 5 where x is the length of the fish in decimeters. What is the length of the fish which weighs 20 pounds? a. Based on the problem, what are you trying to find? b. What is the length of the fish which weighs 20 pounds? 2. Suppose that the safe stopping distance (in meters) of a car is given by , where s represents the speed of the car in miles per hour. Determine the safe speed of the car if it is expected to stop in 50 meters? a. Based on the problem, what are you trying to find? b. Determine the safe speed of the car if it is expected to stop in 50 meters? 3. A banker models the expected value of a newspaper company (in millions) by the polynomial function V(n) = 3n3 + 11n2 – 204n - 20, where n is the number of years in business. If Daily News company has an expected value of Php 120 million, how long was the company operating? a. Based on the problem, what are you trying to find? b. If Daily News company has an expected value of Php 120 million, how long was the company operating? 4. A production company has determined its daily revenue, R, by the polynomial function R(n) = n3 + 25n2 – 200n, where n represents the number of products sold. If the company earns a revenue of 1, 500 pesos on a certain day, how many products was sold? a. Based on the problem, what are you trying to find? Were you surprised that polynomial functions can be applied in real – life problems and have practical uses? What do you need to solve these problems? Enjoy learning as you proceed to the next section.
- 27. 16 b. If the company earns a revenue of 1, 500 pesos on a certain day, how many products was sold? What Is It The previous lesson tells us how to solve problems involving polynomial functions through evaluation. That is, solving for the value of the dependent variable (example of this is P(x), V(x) given the value of the independent variable. Now, we will solve the problems using factoring, but the difference is the problems below are more on solving for the value of the independent variable given the value of the dependent variable. Here, we are going discuss and solve problems together step by step using the lessons you have discussed in the previous days. The problems presented below are things that we could relate to real-life situations Example 1: Suppose a windmill generates power based on the speed of the wind. This is represented by the polynomial function , where s represents the speed of the wind in kilometers per hour. If the windmill generated 50 units of power, what is the speed of the wind? Procedure: Step 1 , P(s) = 50 units First let us rewrite the given. Step 2 Find s when P(s) = 50. Then we identify what is asked on the problem. Step 3 We replace P(s) with 50 since it is given than P(s) = 50. Step 4 We add – 50 to both sides so that the other side is equal to zero. Step 5 0 = (s – 10) (s + 5) We now then solve the equation using factoring. You could use also quadratic equation on solving for the value of s. Step 6 s – 10 = 0 s + 5 = 0 We equate each factor by zero by zero – product property and to solve for the value of s. Step 7 s = 10, s = - 5 By solving each linear equation, we got s = 10 and s = - 5, but we are Excellent! Did you have fun answering each problem? On the next section, let us discuss similar problems step by step to know more about solving problems involving polynomial functions through factoring.
- 28. going to reject – 5 since there is no negative speed of the wind. Thus, we got s = 10. Answer: The speed of the wind should be 10 kilometers per hour to generate 50 units of power. Example 2: A sculptor uses ice blocks to carve the wings of a dragon. The volume, V (in cubic centimeters), of a block of ice can be modeled by the V(t) = t3 + 8t2 – 83t, where t represents the thickness of the block of ice in centimeters. How thick is the block of ice with a volume of 630 cm3? Procedure: Step 1 V(t) = t3 + 8t2 – 83t, V(t) = 630 cm3 First let us rewrite the given. Step 2 Find t when V(t) = 630. Then we identify what is asked on the problem. Step 3 630 = t3 + 8t2 – 83t We replace V(t) with 630 since it is given that V(t) = 630. Step 4 0 = t3 + 8t2 – 83t - 630 We add – 630 to both sides so that the other side is equal to zero. Step 5 0 = (t + 7) (t + 10) (t – 9) We now then solve the equation using factoring. You could also use synthetic division on solving for the value of t. Step 6 t + 7 = 0 t + 10 = 0 t – 9 = 0 We equate each factor by zero by zero – product property and to solve for the value of t. Step 7 t = - 7, t = - 10, t = 9 By solving each linear equation, we got t = -7, t = -10 and t = 9, but we are going to reject – 7 and – 10 since there is no negative measurement for thickness. Thus, we got t = 9. Answer: The block of ice should be 9 centimeters thick to have a volume of 630 cubic centimeters. Great job! Now you are ready for more similar problems. Let us polish more your skills in solving problems involving polynomial functions through factoring.
- 29. 18 19 What’s More Activity 1: Let’s Work on Second Degrees! Direction: Answer the following questions through factoring each given polynomial functions. (3 points each) 1. Careena likes throwing rocks at the ocean. When she throws the rock upward from 160 ft above the ocean, h(t) = - 16t2 + 48t + 160 models the height, h, of the rock above the ocean as a function of time, t, in seconds. How long will it take the rock to reach the ocean (h = 0)? 2. An object moves along the horizontal in a straight line according to the function d = 4t – 2t2, where d is the distance (in meters) and t is the time in seconds. How long will it take the object to travel 70 meters? 3. The weight, w (in kilograms), of a certain fish can be modeled by w = x2 – 7x – 3 where x is the length of the fish in decimeters. What is the length of the fish which weighs 15 kilograms? 4. A certain company can produce a minimum of 10 bicycles up to a maximum of 40 bicycles per week. The profit (in pesos) generated by producing and selling n bicycles per week is given by the formula P(n) = - 5n2 + 400n – 600. How many bicycles must be produced and sold to earn a profit of 5400 pesos in a week? 5. Suppose that the safe stopping distance (in feet) of a car is given by , where s represents the speed of the car in kilometers per hour. Determine the safe speed of the car if it is expected to stop in 40 feet? Activity 2: Let’s Up Another Degree! Direction: Answer the following questions through factoring each given polynomial functions. (3 points each) 1. A banker models the expected value of a company (in millions) by the polynomial function V(n) = n3 – 3n2, where n is the number of years in business. If a production company has an expected value of Php 50 million, how long was the company in business? 2. In colder climates, the cost, C (in dollars), of natural gas to heat homes can vary from one day to the next. This can be represented by the function C(d) = 2d3 – 3d2 – 32d – 3, where d is the number of days that natural gas was used to heat homes. How many days the natural gas was used to heat a certain home which cost them 10 dollars? 3. A box is to be mailed. The volume (in cubic inches) of the box can be expressed as the product of its three dimensions which can be represented by
- 30. 20 the function V(w) = w3 – w2 – 33w – 5, where w is the measure of its width in inches. What is the width of the box if the volume of the box is 58 in3? 4. A production company has determined its daily revenue, R, by the polynomial function R(n) = 6n3 + 150n2 – 1200n, where n represents the number of products sold. If the company earns a revenue of 9 thousand pesos on a certain day, how many products was sold? 5. It has been a while since Christine started working as a part time babysitter and saved her earnings at the bank. The polynomial function E(y) = 100y3 + 400y2 – 1100y represents her savings, with interest, where y is the number of years her savings stayed at the bank. How many years does it need for her savings to reach 3000 dollars with interest? What I Have Learned Direction. Chose a word or phrase from the box below to fill in the missing parts. 1. In solving problems involving polynomial function through factoring, I must first _______ the problem before listing down the given. 2. To create a polynomial function based on the given, I must ______ the given value to its corresponding variables. 3. If I have a polynomial function already based on the given, I must arrange it in ______ form to make it easier to factor. 4. After factoring the arranged polynomial, I must equate each variable factor equal to _____. 5. After equating all factor to zero, it is the time to ______ each linear/ quadratic equation. 6. I must choose the answers which makes ______ or realistic. 7. If the answers include negative values, then I will _____ those values. 8. The values which are _______ are not possible since there is no negative measurement or distance. 9. After getting to choose the answer that makes sense, I must write it in _____ sentence. Good job! You have answered all the questions above. Now, since you are ready, let’s move on to the next section. zero reject sense complete standard solve what is asked negative one substitute what is given negative
- 31. 21 For Items 10 – 15, refer to the problem below. (3 points each). In some parts of the world that experiences cold weather, the cost, C (in dollar), of natural gas to heat homes can vary from one day to the next. This can be modeled by the function C(d) = 2d3 + d2 – 27d – 1, where d is the number of days that natural gas was used to heat homes. 10 – 12. How many days the natural gas was used to heat a certain home which cost them 35 dollars? 13 – 15. Based on your answer on the first question, what answers are to be rejected? Why do you have to reject it? What I Can Do Directions: Read and analyze the situation below. Then, answer the following questions. You own a succulent plant that propagates rapidly. You noticed that it propagates every week and by recording the number of seedlings, you came up with a polynomial function that tells you how many seedlings are there. The polynomial function N(w) = 3w3 + 10w2 – 53w – 20 represents the number of seedlings after w weeks. How many weeks are needed for the succulents to propagate up to 120 seedlings? Answer the following: a. Based on the problem, what are you trying to find? b. How many weeks are needed for the succulents to propagate up to 120 seedlings?
- 32. 22 Assessment Let us find out how much you have learned on this lesson. Answer all items. Choose the letter of the correct answer to each question. Write the chosen letter on a separate sheet of paper. For Items number 1 – 5, Refer to the problem below. Lolita created a machine that helped a lot of people in a certain province. Due to her invention, the businesses that uses her invention started to boom due to the low cost of operation and maintenance using the machine. The polynomial function P(x) = 12x3 - 85x2 – 86x – 1 represents the profit (in thousands of pesos) of a certain company that uses the machine for production where x is the number of products produced using the machine in thousands. How many products are produced when the company profited 15 thousand pesos? 1. What is asked on the problem? A. How many products are produced when the company profited 15 thousand pesos. B. Find P(x) when x is equal to 15 thousand pesos. C. Solve for the profit when 15 thousand products are processed. D. Find the number of products processed. 2. Based on the problem, P(x) represents __________. A. the products are produced when the company profited 15 thousand pesos. B. the value P(x) when x is equal to 15 thousand pesos. C. the profit of a certain company that uses the machine for production. D. the number of products processed. 3. What is the degree of the polynomial presented on the problem? A. 2nd B. 3rd C. 4th D. 5th 4. How many products are produced when the company profited 15 thousand pesos. A. 5 000 B. 6 500 C. 8 000 D. 9 500 5. What number should be substituted to P(x) based on the problem? A. 15 B. 150 C. 1, 500 D. 15, 000 For Items number 6 – 10, Refer to the problem below. The daily revenue, R (in thousands of pesos), of sales company is modeled by the polynomial function R(d) = 2d3 + 11d2 – 16d - 5, where d represents the number of deals closed. If the company earns a revenue of 100, 000 pesos on a certain day, how many closed deals were made? 6. Based on the problem, d represents __________. A. the polynomial function used to compute the revenue. B. the revenue of a certain day which is 100, 000 pesos. C. the daily revenue of the sales company. D. the number of deals closed.
- 33. 23 7. The polynomial function represents ____________. A. the polynomial function used to compute the revenue. B. the revenue of a certain day which is 100, 000 pesos. C. the daily revenue of the sales company. D. the number of deals closed. 8. If the company earns a revenue of 100, 000 pesos on a certain day, how many closed deals were made? A. 3 B. 4 C. 5 D. 6 9. What number should be substituted to P(x) based on the problem? A. 100 B. 1, 000 C. 10, 000 D. 100, 000 10. Based on the problem, R(d) represents __________. A. the polynomial function used to compute the revenue. B. the revenue of a certain day which is 100, 000 pesos. C. the daily revenue of the sales company. D. the number of deals closed. For Items number 11 – 15, Refer to the problem below. A sculptor uses ice blocks to carve his masterpiece. The volume, V (in cubic centimeters), of a block of ice can be modeled by the V(t) = 4t3 - 15t2 – 81t - 5, where t represents the thickness of the block of ice in centimeters. How thick is the block of ice with a volume of 65 cm3? 11. Based on the problem, V(t) represents __________. A. the thickness of a block of ice in centimeters. B. the thickness of a block of ice which is 65 cm3. C. the dimensions of a block of ice. D. the volume of a block of ice. 12. How thick is the block of ice with a volume of 65 cm3? A. 6 cm B. 7 cm C. 8 cm D. 9 cm 13. The polynomial function represents ____________. A. the thickness of a block of ice in centimeters. B. the thickness of a block of ice which is 65 cm3. C. the dimensions of a block of ice. D. the volume of a block of ice. 14. Based on the problem, t represents __________. A. the thickness of a block of ice in centimeters. B. the thickness of a block of ice which is 65 cm3. C. the dimensions of a block of ice. D. the volume of a block of ice. 15. What is the degree of the polynomial presented on the problem? A. 2nd B. 3rd C. 4th D. 5th
- 34. 24 Additional Activities Activity 3: Missing You! Directions: Fill up the missing parts of the solution to the problem. Copy and answer on a separate sheet of paper. Alice was told that the volume of a rectangular block is modeled by the function V(x) = 14x3 - 57x2 – 419x - 3, where x is the length in meters of a rectangular block. How long is a rectangular block if it has a volume of 165 m3? Procedure: Step 1 1. _____________ 2. _____________ First let us rewrite the given. Step 2 3. _____________ Then we identify what is asked on the problem. Step 3 4. _______________ 5. We replace V(x) with ___ since it is given that V(x) = ___. Step 4 0 = 14x3 - 57x2 – 419x - 168 6. We add ___ to both sides so that the other side is equal to zero. Step 5 7 – 9. 0 = (_____) (______) (______) We now then solve the equation using factoring. You could also use synthetic division on solving for the value of x. Step 6 10. ______ = 0 ______ = 0 ______ = 0 We equate each factor by zero by zero – product property and to solve for the value of x. Step 7 x = , x = , x = 8 11 – 13. By solving each linear equation, we got x = ______, x = ____ and x = ____, but we are going to reject ___ and ____ since there is no negative measurement for length. Thus, we got x = ___. 14 – 15. Answer: ______________________________________.
- 35. 25 Activity 4: Ref! Substitute! Me Again! Directions: Answer the following question through evaluating each given polynomial function. (3 points each) 1. The weight, w (in kilograms), of a wild boar can be modeled by w = 3x2 – 21x – 9 where x is the length of the wild boar in decimeters. What is the length of the wild boar which weighs 45 kilograms? 2. An object moves along the horizontal in a straight line according to the function d = 2t2 – 5t, where d is the distance (in meters) and t is the time in seconds. How long will it take the object to travel 42 meters? 3. A banker models the expected value of a company (in millions) by the polynomial function V(n) = 5n3 – 15n2, where n is the number of years in business. If a production company has an expected value of Php 250 million, how long was the company in business? 4. A drugstore that sells a certain brand of Vitamin C tablets predicts that their profit, P (in millions of pesos), can be modeled by the polynomial function P(a) = a3 + 2a2 – 5a - 1, where a is the amount spent on advertising (in thousands of pesos). How much did they spent on advertising if they profited 5 million pesos? 5. In 2010, the population P of a certain municipality is modeled by the function P(t) = 100t3 + 200t2 - 4300t where t is the number of years after 2010. On what year will the population of the municipality have reached 14, 000 citizens?
- 36. 26 Lesson 3 Modeling Polynomials (Day 3 and 4) What I Need to Know After going through this lesson, you are expected to: 1. identify the necessary information to be used in solving real-life problems involving polynomial functions; and 2. apply polynomial functions to real-life problems and produce a concrete object that satisfies the condition given on the problem. What I Know Let us find out how much you already know about the topic on this lesson. Answer all items. Choose the letter of the correct answer to each question. Write the chosen letter on a separate sheet of paper. For Items number 1 – 5, Refer to the problem below. A steel company was hired to build a rectangular container for shipping. The client wants the height of the pool to be 1 meter more than twice the width and the length is 3 meters less than five times the width. The client also specified that the container must have a capacity of 252 cubic meters. 1. If w represents the measurement of the width, which of the following mathematical model satisfy the condition of the above problem? A. (2w - 3) (5w + 1) (w) = 252 C. (5w - 3) (2w + 1) (w) = 252 B. (w + 1) (w - 3) (w) = 252 D. (5w - 2) (3w + 1) (w) = 252 2. How wide is the container? A. 3 meters B. 4 meters C. 5 meters D. 6 meters 3. How high is the container? A. 4 meters B. 5 meters C. 6 meters D. 7 meters 4. How long is the container? A. 12 meters B. 14 meters C. 16 meters D. 18 meters 5. What are the dimensions of the swimming pool? (Follow V = LWH). A. 17 m by 5 m by 5 m C. 12 m by 5 m by 8 m B. 12 m by 3 m by 7 m D. 17 m by 7 m by 5 m
- 37. 27 For Items number 6 – 10, Refer to the problem below. A rectangular box has a width of w feet. The height is 5 feet less than the width. The length is 1 foot more than twice the width. If the volume of the box is 24 times the measure of the length. What are the dimensions of the box? 6. If w represents the measurement of the width, which of the following mathematical model satisfy the condition of the above problem? A. (2w + 1) (w - 5) (w) = 24 (2w + 1) B. (2w - 1) (w - 5) (w) = 24 (2w - 1) C. (2w + 1) (w - 2) (w) = 24 (2w + 1) D. (2w + 1) (w - 1) (w) = 24 (2w + 1) 7. How wide is the box? A. 6 feet B. 7 feet C. 8 feet D. 9 feet 8. How high is the box? A. 3 feet B. 4 feet C. 5 feet D. 6 feet 9. How long is the container? A. 16 feet B. 17 feet C. 18 feet D. 19 feet 10. What are the dimensions of the rectangular box? (Follow V = LWH). A. 17 ft by 8 ft by 3 ft C. 12 ft by 7 ft by 8 ft B. 12 ft by 3 ft by 7 ft D. 17 ft by 8 ft by 5 ft For Items number 11 – 15, Refer to the problem below. A construction firm is asked to construct a rectangular swimming pool. The width of the pool is 6 feet more than the depth, and the length is 10 feet more than the depth. The pool holds 825 cubic feet of water. What are the dimensions of the pool? 11. If d represents the measurement of the depth, which of the following mathematical model satisfy the condition of the above problem? A. (d - 6) (d - 10) (d) = 825 B. (d + 6) (d + 10) (d) = 825 C. (d - 6) (d + 10) (d) = 825 D. (d + 6) (d - 10) (d) = 825 12. How wide is the swimming pool? A. 7 feet B. 9 feet C. 11 feet D. 13 feet 13. How deep is the swimming pool? A. 3 feet B. 4 feet C. 5 feet D. 6 feet 14. How long is the swimming pool? A. 15 feet B. 16 feet C. 17 feet D. 18 feet 15. What are the dimensions of the rectangular box? (Follow V = LWH). A. 15 ft by 11 ft by 5 ft C. 15 ft by 11 ft by 8 ft B. 12 ft by 11 ft by 7 ft D. 17 ft by 8 ft by 5 ft
- 38. 28 What’s In Factor each polynomials function then solve for its value. 1. f(x) = 3x2 – 4x + 7 2. H(s) = 2s3 + 13s2 + 26s + 15. 3. V(x) = x3 + 5x2 – 4x - 20. What’s New Let’s Explore Nana is planning to make a sculpture for her parents. She has a rectangular block of wood, with a dimension 3ft x 4ft x 5ft, that she wants to reduce in size by shaving off the same amount from the length, width, and height. She wants to reduce the volume of the wood block into 24 cubic feet. What are the dimensions of the new wood block? Questions/ Tasks: 1. Based on the problem, create an expression that will represents the new wood block dimensions. a. width: _______ b. length: ________ c. height: ______ 2. Create a mathematical model that represents the volume of the new wood block. (Hint: V = lwh). 3. What are the dimensions of the new wood block? a. width: _______ b. length: ________ c. height: ______ Were you able to apply all the necessary concepts in factoring polynomial function? The next activity will let you see another way of solving problems involving polynomial functions.
- 39. 29 Let’s try this Answer the following: 1. A rectangular box has a width of w centimeters. The height is 2 centimeters less than the width. The length is 4 centimeters more than twice the width. If the volume of the box is 8 times the measure of the length. a. Construct a mathematical equation that models the given from the problem. b. What are the dimensions of the box? 2. A rectangular solid container used in shipping goods has a volume of 84 cubic meters. A client made a special request to the manufacturer that the length of the container must be one meter longer than the width, and the height must be one meter longer than twice the width. a. Construct a mathematical equation that models the given from the problem. b. What are the dimensions of the container? 3. A big balikbayan box has a volume of 264 cubic feet. Its length is two feet longer than the width, and the height is three feet longer than twice the width. a. Construct a mathematical equation that models the given from the problem. b. What are the dimensions of the box? What Is It The previous lesson tells us how to solve problems involving polynomial functions specifically on finding the maximum area given the perimeter. We first constructed a mathematical model then solve it to determine the length and the width. Now, we will solve similar problems, but the difference is the problems now are all 3-dimensional objects which tells us that aside from the length and width, we now got the height. Here, we are going discuss and solve problems together step by step using the lessons you have discussed in the previous days. The problems presented below are things that we could relate to real-life situations Were you surprised that polynomial functions can be applied in real – life problems and have practical uses? What do you need to solve these problems? Enjoy learning as you proceed to the next section.
- 40. 30 Example 1: Mitzi is planning to make an ice sculpture for her parents. She has a rectangular block of ice, with a dimension 3ft x 4ft x 5ft, that she wants to reduce in size by shaving off the same amount from the length, width, and height. She wants to reduce the volume of the ice block into 24 cubic feet. What are the dimensions of the new ice block? Procedure: Step 1 3ft x 4ft x 5ft block of ice, 24 cubic feet as the volume of the new ice block First let us rewrite the given. Step 2 What are dimensions of the new ice block? Identify what is asked on the problem. Step 3 x = amount of ice to be shaved off We let x as the amount of ice to be shaved off since we want to shave off the same amount from its length, width, and height. Step 4 (3 – x), (4 – x), (5 – x) We subtract x to each of the original dimension of the ice block which is 3ft x 4ft x 5ft since it was stated on the problem. Step 5 (3 – x) (4 – x) (5 – x) = 24 We are now creating our mathematical model by multiplying all the measurements of the dimensions. You might be asking why we multiplied the new dimensions, remember that we dealing with the volume of a rectangular solid which is Volume = length x width x height. Also, we equate it with 24 since it is stated that the volume of the new ice block is 24 cubic feet. Step 6 (12 – 7x + x2) (5 – x) = 24 Then we simply the polynomial, we first multiplying (3 – x) by (4 – x) which resulted in 12 – 7x + x2. You could also multiply the other terms first since multiplication is commutative. Step 7 60 – 47x + 12x2 – x3 = 24 Next is we multiply (12 – 7x + x2) by (5 – x) which is equal to 60 – 47x + 12x2 – x3. Step 8 36 – 47x + 12x2 – x3 = 0 Let us add – 24 to both sides so that the other side will be equal to zero. Step 9 – x3 + 12x2 – 47x + 36 = 0 Let us arrange the polynomial in standard form so that it would be easier to factor later on. Step 10 x3 - 12x2 + 47x - 36 = 0 We are going to multiply -1 to both sides so that the leading term is positive and easier way when factoring the
- 41. 31 polynomial. Step 11 (x – 1) (x2 – 11x + 36) = 0 By factoring the polynomial, we got the following. You could also use synthetic division in finding the factor of the polynomial. Step 12 (x – 1) = 0, (x2 – 11x + 36) = 0 We set each factor equal to zero by zero product property. Step 13 x = 1, By solving for x, we get these numbers. But is not a real number, thus our answer is x = 1. On the case of (x2 – 11x + 36) = 0, we used the quadratic formula to solve for the value of x since it is not factorable. Step 14 2ft by 3ft by 5 ft By substituting x = 1 on (3 – x), (4 – x), and (5 – x), we now have the new dimension of the ice block. Answer: The dimensions of the new ice block is 2ft by 3ft by 5 ft. Example 4: A rectangular box has a width of w centimeters. The height is 2 centimeters less than the width. The length is 4 centimeters more than twice the width. If the volume of the box is 8 times the measure of the length. What are the dimensions of the box? Procedure: Step 1 width = w height = w – 2 length = 2w + 4 volume = 8 (2w + 4) First let us rewrite the given. Step 2 What are dimensions of the prism Identify what is asked on the problem. Step 3 (w) (w - 2)(2w + 4) = 8 (2w + 4) We are now creating our mathematical model by multiplying all the measurements of the dimensions. You might be asking why we multiplied the new dimensions, remember that we dealing with the volume of a rectangular prism which is Volume = length x width x height. Also, we equate it with 8 (2w + 4) since it is stated that the volume of the rectangular box is 8 times the measure of the length.
- 42. 32 Step 4 (w) (w - 2) (2w + 4) = 16w + 32 We first simplify 8 (2w + 4) by distributive property of multiplication. Step 5 (w2 – 2w) (2w + 4) = 16w + 32 Then we simply the polynomial, we first multiplying w by (w - 2) which resulted in w2 – 2w. You could also multiply the other terms first since multiplication is commutative. Step 6 2w3 - 8w = 16w + 32 Next is we multiply (w2 – 2w) by (2w + 4) which is equal to 2w3 - 8w. Step 7 2w3 – 24w - 32 = 0 Let us add – 16w - 32 to both sides so that the other side will be equal to zero. Step 8 w3 – 12w - 16 = 0 We multiply ½ to both sides so that the leading term have a coefficient of 1 and to make it easier on factoring the polynomial later. Step 9 (w + 2) (w + 2) (w – 4) = 0 By factoring the polynomial, we got the following. You could also use synthetic division in finding the factor of the polynomial. Step 10 (w + 2) = 0, (w + 2) = 0, (w – 4) = 0 We set each factor equal to zero by zero product property Step 11 w = - 2, -2, 4 By solving for w, we get these numbers, but we are talking about measurement thus we reject w = - 2 since there is no negative measurement. Hence, our answer is w = 4 Step 12 width = 4 cm height = 2 cm length = 12 cm By substituting w = 4 on width = w, height = w – 2, and length = 2w + 4, we got the dimension of the box. Answer: The dimensions of the box is 12 cm by 4 cm by 2 cm. (Following LxWxH) Great job! Now you are ready for more similar problems. Let us polish more your skills in solving problems involving polynomial functions by making models.
- 43. 33 What’s More Activity 1: RACk my Head! (Read, Analyze, and Create) Direction: Read and analyze each problem then create a mathematical model for that satisfy the condition on each problem. (3 points each) 1. A construction firm is asked to construct a rectangular swimming pool. The width of the pool is 5 feet more than the depth, and the length is 35 feet more than the depth. The pool holds 2000 cubic feet of water. 2. On a certain ancient ruin, an archaeologist discovered a huge concrete block with a volume of 945 cubic meters. The dimensions of the block are x meters high by 12x – 15 meters long by 12x – 21 meters wide. 3. A sculptor plans to make a sculpture on the shape of a rectangular prism. He wants the height and the width to be 5 inches less than the length. What should be the dimension of the prism if he wants to use a 250 cubic inches clay? 4. Find the length of the edge of a cube if its length is increased by 6 cm, its width is increased by 12 cm and its height is decreased by 4 cm results in double its volume. 5. A rectangular box has a dimension of 12 inches long, 4 inches wide, and 4 inches high. If the length and the width is decreased and the height is increased by the same amount, a second box is formed, and its volume is that of the original box. Activity 2: Find Me! Direction: Using the mathematical models you have created on Activity 1: RACk my Head, solve each problem as stated. (3 points each) 1. A construction firm is asked to construct a rectangular swimming pool. The width of the pool is 5 feet more than the depth, and the length is 35 feet more than the depth. The pool holds 2000 cubic feet of water. What are the dimensions of the pool? 2. On a certain ancient ruin, an archaeologist discovered a huge concrete block with a volume of 945 cubic meters. The dimensions of the block are x meters high by 12x – 15 meters long by 12x – 21 meters wide. What is the height of the block? 3. A sculptor plans to make a sculpture on the shape of a rectangular prism. He wants the height and the width to be 5 inches less than the length. What should be the dimension of the prism if he wants to use a 250 cubic inches clay? 4. Find the length of the edge of a cube if its length is increased by 6 cm, its width is increased by 12 cm and its height is decreased by 4 cm results in double its volume.
- 44. 34 5. A rectangular box has a dimension of 12 inches long, 4 inches wide, and 4 inches high. If the length and the width is decreased and the height is increased by the same amount, a second box is formed, and its volume is that of the original box. Find the dimension of the second box. What I Have Learned Direction: Chose a word or phrase from the box below to fill in the missing parts. 1. In solving problems involving polynomial function, I must first ______ the problem before listing down the given. 2. One of the effective ways to not forget the given is to _____ it down first. 3. Before solving the problem, I must identify first __________. 4. To evaluate a polynomial function, I would just simply ______ the given value to its corresponding variables then solve. 5. To get the volume of a rectangular prism, I must multiply its _____, _____, and ______. 6. After knowing what is asked, I must create a __________ before I can start solving the problem. 7. To make it easier on factoring polynomials, it is advised to _______ the polynomials into its standard form. 8. After factoring the arranged polynomial, I must equate each variable factor equal to _____. 9. After equating all factor to zero, it is the time to ______ each linear/ quadratic equation. 10. I must _____ the negative values since there is no negative measurement. 11. If I get multiple values of x, I will only consider ______ and ______ values. 12. After getting the exact value, my answer should be in ______ sentence. Good job! You have answered all the questions above. Now, since you are ready, let’s move on to the next section. width mathematical model read height complete write length what is asked positive substitute zero solve reject arrange real imaginary
- 45. 35 For Items 13 – 15, refer to the problem below. (3 points). A sculptor plans to make a sculpture on the shape of a rectangular prism. He wants the height and the width to be 3 inches less than the length. What should be the dimension of the prism if he wants to use a 112 cubic inches clay. What I Can Do Directions: Read and analyze the situation below. Then, answer the following questions. There is a contest on your school for a themed school dance. Your class section came up with an icy themed dance inspired by the movie “Frozen”. You and your team were assigned to create props for the said contest, and you came up with a pyramid ice sculpture idea. So, you are to create an ice mold in a shape of a pyramid with a height of 2 feet greater than the length of each side of its square base. The volume of the pyramid should be 15 cubic feet. Perform the following: a. Create a mathematical model that satisfy the conditions given on the problem above. (Hint: , where a = edge of the base of the pyramid and h = height of the pyramid) b. What are the dimensions of the pyramid? c. Suppose you increase all the dimensions by 1 foot each. How big is capacity of the new pyramid? Compared to the original pyramid, which is bigger? By How much?
- 46. 36 Assessment Let us find out how much you’ve learned about the topic on this lesson. Answer all items. Choose the letter of the correct answer to each question. Write the chosen letter on a separate sheet of paper. For Items number 1 – 5, refer to the problem below. A factory needs a box with a capacity of 1728 cubic inches. The length of the box should be 6 inches greater than the height and the width should be 4 inches less than the height. What are the dimensions of the box? 1. If h represents the measurement of the height, which of the following mathematical model satisfy the condition of the above problem? A. (h + 6) (h + 4) (h) = 1728 C. (h + 6) (h - 4) (h) = 1728 B. (h - 6) (h - 4) (h) = 1728 D. (h - 6) (h + 4) (h) = 1728 2. How wide is the box? A. 4 inches B. 6 inches C. 8 inches D. 10 inches 3. How high is the box? A. 6 inches B. 8 inches C. 10 inches D. 12 inches 4. How long is the box? A. 12 inches B. 14 inches C. 16 inches D. 18 inches 5. What are the dimensions of the box? (Follow V = LWH). A. 17 in by 12 in by 5 in C. 18 in by 8 in by 12 in B. 18 in by 3 in by 7 in D. 17 in by 7 in by 5 in For Items number 6 – 10, refer to the problem below. A candy factory needs a box with a volume of 420 cubic inches. The height should be 2 inches less than the width and the length is 5 inches greater than the width. What are the dimensions of the box? 6. If w represents the measurement of the width, which of the following mathematical model satisfy the condition of the above problem? A. (w - 2) (w + 5) (w) = 420 C. (w + 2) (w - 5) (w) = 420 B. (w + 2) (w + 5) (w) = 420 D. (w - 2) (w - 5) (w) = 420 7. How wide is the box? A. 3 inches B. 5 inches C. 7 inches D. 9 inches 8. How high is the box? A. 5 inches B. 6 inches C. 7 inches D. 8 inches 9. How long is the box? A. 12 inches B. 14 inches C. 16 inches D. 18 inches 10. What are the dimensions of the box? (Follow V = LWH). A. 15 in by 12 in by 5 in C. 18 in by 8 in by 12 in B. 12 in by 3 in by 7 in D. 12 in by 7 in by 5 in
- 47. 37 For Items number 11 – 15, refer to the problem below. In 1980, archaeologists at the ruins of Caesara discovered a huge hydraulic concrete block with a volume of 1638 cubic feet. The block’s dimension is x feet wide, 3x + 5 long, and 2x – 5 high. Find the dimensions of the block. 11. If x represents the measurement of the width, which of the following mathematical model satisfy the condition of the above problem? A. (3x + 6) (2x - 5) (x) = 1638 B. (3x + 5) (2x - 5) (x) = 1638 C. (3x - 5) (2x - 5) (x) = 1638 D. (3x - 6) (2x + 5) (x) = 1638 12. How wide is the hydraulic concrete block? A. 7 feet B. 9 feet C. 11 feet D. 13 feet 13. How high is the hydraulic concrete block? A. 3 feet B. 6 feet C. 9 feet D. 12 feet 14. How long is the hydraulic concrete block? A. 11 feet B. 16 feet C. 21 feet D. 26 feet 15. What are the dimensions of the hydraulic concrete block? (Follow V = LWH). A. 26 ft by 7 ft by 5 ft C. 21 ft by 11 ft by 8 ft B. 26 ft by 7 ft by 9 ft D. 16 ft by 9 ft by 13 ft Additional Activities Activity 3: Put It in The Parts! Directions: Fill up the missing parts of the solution to the problem. Copy and answer on a separate sheet of paper. A candy factory needs a box with a volume of 308 cubic inches. The height should be 5 inches less than the thrice the width and the length is 3 inches greater than the twice the width. What are the dimensions of the box? Procedure: Step 1 1. _____________ 2. _____________ 3. _____________ 4. _____________ First let us rewrite the given. Step 2 What are the dimensions of the box? Identify what is asked on the problem. Step 3 5. _______________ We are now creating our mathematical model by multiplying all the
- 48. 38 measurements of the dimensions. You might be asking why we multiplied the new dimensions, remember that we dealing with the volume of a rectangular prism which is Volume = length x width x height. Step 4 6. (________) (2w + 3) = 308 Let us simply the polynomial, we first multiplying w by (3w - 5) which resulted in _______. Step 5 6w3 – w2 – 15w = 308 We multiply 3w2 – 5w by 2w + 3 which resulted in 6w3 – w2 – 15w. Step 6 7. _____________ 8. Let us add _____ to both sides so that the other side will be equal to zero. Step 7 9. (______) (_________) = 0 By factoring the polynomial, we got the following. You could also use synthetic division in finding the factor of the polynomial. Step 8 10. ______ = 0 11. ______ = 0 We set each factor equal to zero by zero product property Step 9 12. w = ____ 13. w = ± ________ By solving for w, we get these numbers. We will reject the other 2 answers since they are imaginary numbers. Step 10 14. __________ By substituting w = 4 on width = w, height = 3w – 5, and length = 2w + 3, we got the dimension of the box. 15. Answer: ______________________________________. Activity 4: Creating Models! Direction: Read and analyze each problem then create a mathematical model for that satisfy the condition on each problem. (3 points each) 1. A factory needs a box with a capacity of 1320 cubic inches. The length of the box should be 7 inches greater than twice the height and the width should be 4 inches less than the height. 2. A sculptor plans to make a sculpture on the shape of a rectangular prism. He wants the height and the width to be 10 inches less than the length. 3. On a certain ancient ruin, an archaeologist discovered a huge concrete block with a volume of 1935 cubic meters. The dimensions of the block are x meters high by 2x – 15 meters long by 5x – 21 meters wide. 4. A rectangular solid container used in shipping goods has a volume of 65 cubic meters. A client made a special request to the manufacturer that the length of the container must be five meters longer than twice the width, and the height must be one meter shorter than the width.
- 49. 39 5. A big balikbayan box has a volume of 520 cubic feet. Its length is two feet longer than thrice the width, and the height is three feet shorter than twice the width. Summary • On solving word problems involving polynomials, we must consider first these steps. These are not just applicable in solving word problems involving polynomials but also to any word problems. a. How to use a problem-solving strategy to solve word problems. i. Read the problem. Make sure all the words are understood. ii. Identify what the give and what we are looking for. iii. Name what we are looking for. Choose a variable to represent quantity. iv. Translate into mathematical model. It may be helpful to restate the problem in one sentence will all the important information. Then, translate the English sentence in to equation. v. Solve the equation using appropriate algebra techniques. vi. Check the answer in the problem and make sure it makes sense. vii. Answer the question with a complete sentence and label your answers.
- 50. 40 Assessment: (Post-Test) Directions: Choose the letter of the correct answer. Write the chosen letter on a separate sheet of paper. 1. A sofa set dealer determines that its profit, P, in thousands of pesos, can be modeled by the function P(x) = 0.0012x3 + 2x – 25, where x represents the number of sofa set sold. What is the profit when x = 20? A. Php 24, 600 B. Php 25, 300 C. Php 26, 700 D. Php 27, 900 2. In this school year, 100 high school students participated in one-week Human vs Zombie game. The number of zombies, Z, after d days of playing can be modeled by the function Z(d) = 5d3 – 3d2 + 15. How many days did it take for the number of zombies to reach 43? A. 1 B. 2 C.3 D. 4 3. Edd is going to throw a rock from the top of a trail overlooking the ocean. When he throws the rock upward from 160 ft above the ocean, the function h(t) = -16t2 + 48t + 160 models the height, h, of the rock above the ocean as a function of time, t. Find the height of the rock at t = 2 seconds. A. 148 ft B. 164 ft C. 180 ft D. 192 ft 4. The estimated number of electric motorcycles, V (in thousands), in the Philippines can be modeled by the polynomial function: V(y) = 0.15128y3 – 3.28234y2 + 23.7565y – 2.041 where y represents the number of years after the year 2005. Find the estimated number of electric motorcycles at the end of the year 2008. Round up to the nearest thousands. A. 42, 000 B. 43, 000 C. 44, 000 D. 45, 000 5. The weight, w (in kilograms), of a certain crab can be modeled by w = 0.00304x3, where x is the length of the arm span of the crab in inches. What is the weight of a crab with an arm span of 12 inch? A. 5.00 kg B. 5.25 kg C. 5.50 kg D. 5.75 kg 6. The profit, P (in millions of pesos), for a flash drive manufacturer can be modeled by P = n3 + 2n2 - 23n where n (in millions) is the number of flash drive produced. How many flash drives was produced if the profit reached 60 million pesos? A. 3, 000, 000 B. 4, 000, 000 C. 5, 000, 000 D. 6, 000, 000 7. The volume, V (in cubic feet), of a rectangular wood block is modeled by the polynomial function V(w) = 12w3 – 38w2 – 104w where w is the width (in feet) of the wood block. What is the width of the wood block if its volume is 30 cubic feet? A. 4 feet B. 5 feet C. 6 feet D. 7 feet 8. During a 10-year period, the amount (in millions of pesos) of school supplies (S) sold can be modeled by S(t) = - 10t3 + 25t2 – 140t + 332, where t is in years. Find the amount of school supplies sold in 3 years. A. Php 794, 000, 000 B. Php 796, 000, 000 C. Php 795, 000, 000 D. Php 797, 000, 000
- 51. 41 For Item number 9 – 11, refer to the problem below. On a glass factory, paperweights are created by pouring molten glass into molds. Each mold is a rectangular prism with a height 5 cm greater than the length of each side of its square base. Each mold holds 1008 cubic centimeters of glass. 9. If x represents the length of each side of the square base, which of the following mathematical model satisfy the condition of the above problem? A. (x + 5) (x) = 1008 B. (x + 5) (x) (x) = 1008 C. (x + 5) (x + 5) (x) = 1008 D. 1008 (x + 5) = (x) (x) 10. What is the height of the glass mold? A. 5 cm B. 7 cm C. 12 cm D. 16 cm 11. What are the dimensions of the glass molds? (Follow V = LWH) A. 7 cm by 5 cm by 12 cm B. 7 cm by 12 cm by 12 cm C. 7 cm by 7 cm by 12 cm D. 7 cm by 8 cm by 8 cm For Item number 12 – 15, refer to the problem below. A construction company was hired to build a swimming pool for an attraction and high lights in a wedding venue. The client wants the width of the pool to be 2 meters less than the depth and the length is 10 meters more than the depth. The client also specified that the pool must have a water capacity of 595 cubic meters. 12. If d represents the measurement of the depth, which of the following mathematical model satisfy the condition of the above problem? A. (d - 2) (d + 2) (d) + 10 = 595 B. (d - 2) (d - 10) (d) = 595 C. (d - 10) (d + 2) (d) = 595 D. (d - 2) (d + 10) (d) = 595 13. How deep is the pool? A. 1 meters B. 3 meters C. 5 meters D. 7 meters 14. How long is the pool? A. 15 meters B. 17 meters C. 19 meters D. 21 meters 15. What are the dimensions of the swimming pool? (Follow V = LWH) A. 17 m by 5 m by 7 m B. 12 m by 2 m by 7 m C. 12 m by 5 m by 7 m D. 17 m by 8 m by 5 m
- 52. Lesson 1 What In Number Lea ding ter m End Behavior Y intercept Number of turning points 1 X8 Up – left, Up – right 9 7 2 X7 Down – left, Up – right 56 6 3 -X3 Down – right, Up - left 0 2 Lesson 1 What’s New 1A. 15 wild boars 1B. They are not the same since if you use the polynomial function the answer is with decimal which is not realistic because boars are counted as a whole. 1C. Since we don’t have restrictions and not considering other variables that might happen, we can predict the number of wild boars in the 10th year Lesson 1 Let’s Try this 1 a. 6425 b. 9900 2 a. 208.9L b. 196.2L 3 a. P 33, 500 b. P 269, 500 Lesson 1 What’s More Activity 1 1. 30.4 carats 2. Php 407 600 3. 4, 714, 000 4. 594 eggs 5. 35 balls Lesson 1 What’s More Activity 2 1. P 1, 380, 000, 000 2. 71 hexagons 3. 132, 674 residents 4. 1000 units 5. 7293 cable TV systems Lesson 1 What I Know 1 A 6 C 11 B 2 A 7 A 12 D 3 D 8 D 13 A 4 B 9 A 14 D 5 D 10 D 15 C Lesson 1 What I Have Learned 1. Read 2. Write 3. What is Asked 4. Substitute 5. Mathematical Model 6. P 7, 500 7. P 14, 250 8. P 37, 750 9. P 149, 000 10. Remove all the pollutants since if I do it by percent it would be very costly in the long run Lesson 1 What Can I Do a. 5, 715, 000, 000 laptops b. 41, 750, 000, 000 pesos c. After evaluating the polynomial when n = 5, I’ve multiplied the result by 10 since 1 laptop sold gets 10 pesos donation. Lesson 1 Assessment 1 C 6 C 11 B 2 A 7 A 12 D 3 A 8 C 13 A 4 A 9 D 14 B 5 B 10 D 15 A 42 Key to Answers
- 53. Lesson 1 Additional Act. 3 1 – 2. x = 2, V(x) = 5x6 – 2x5 + 7x4 – 10x3 + 14x2 – 8x + 8 3. Find the volume of a certain block with a length of 2 meters. 4. V (2) = 5(2)6 – 2(2)5 + 7(2)4 – 10(2)3 + 14(2)2 – 8(2) + 8 5. Substitute x with 2 6. V (2) = 5(64) – 2(32) + 7(16) – 10(8) + 14(4) – 8(2) + 8 7. V (2) = 320 - 64 + 112 – 80 + 56 – 16 + 8 8. V (2) = 336 9 – 10. The volume of the block is 336 m3 when its length is 2 meters. Lesson 1 Additional Act. 4 1. P 188, 000 2. 14 eggs 3. P 25, 000, 000 4. 141 hexagons 5. 22, 250 residents Lesson 2 What In 1. 19 2. 47 3. 1510 Lesson 2 What’s New 1. P 13, 000 2. 7 sets 3. The money you’ve earned by selling sets of plants 4. 42 since the problem states that E(p) is in hundreds of pesos Lesson 2 Let’s Try this 1 a. What is the length of the fish which weighs 20 pounds? b. 5 decimeters 2 a. Determine the safe speed of the car if it is expected to stop in 5o meters. b. 7 miles per hour 3 a. How long was the daily news operating if it is expected value is 120 million? b. 7 years 4. a. how many products were sold if the company earns 1500. b. 10 products Lesson 2 What’s More Activity 1 1. 5 seconds 2. 7 seconds 3. 9 decimeters 4. 20 bicycles 5. 20 kph Lesson 2 What I Know 1 B 6 C 11 D 2 C 7 B 12 C 3 D 8 B 13 D 4 A 9 C 14 A 5 B 10 B 15 B Lesson 2 What I Have Learned 1. Read 2. Substitute 3. Standard 4. Zero 5. Solve 6. Sense 7. Reject 8. Negative 9. Complete 10 – 12. 4 days 13 – 15. d = -3, d = -3/2 since there is no negative number of days. Lesson 2 What’s More Activity 2 1. 5 years 2. A 3. 7 inches 4. 10 products 5. 3 years 43
- 54. Lesson 2 What Can I Do a. the number of weeks needed for the succulents to propagate up to 120 seedlings. b. 4 weeks Lesson 2 Assessment 1 A 6 D 11 D 2 C 7 C 12 B 3 B 8 A 13 D 4 C 9 A 14 A 5 A 10 C 15 B Lesson 2 Additional Act. 3 1 – 2. V(x) = 14x3 - 57x2 – 419x – 3, V(x) = 165 m3 3. How long is a rectangular block if it has a volume of 165 m3 4. 165 = 14x3 - 57x2 – 419x – 3 5. V(x) = 165 6. - 165 7 – 9. 0 = (x – 8) (2x – 7) (7x – 3) 10. x – 8 = 0, 2x – 7 = 0, 7x – 3 = 0 11 - 13. x = 8, x = -3/7, x = -7/2 -3/ 7, -7/2, 8 14 – 15. The length of the rectangular block is 8 m if the volume is 165 m3. Lesson 2 Additional Act. 4 1. 9 decimeters 2. 6 seconds 3. 5 years 4. P 2 000 5. 7 years Lesson 3 What In 1. f(x) = (3x – 7)(x + 1); x = 7/3, x = -1 2. H(s)=(x +1)(x + 3)(2x + 5); x = -1, -3, -5/2 3. V(x)=(x + 5)(x + 2)(x - 2); x = -5, 2, -2 Lesson 3 What’s New 1. a. 3 – x, b. 4 – x, c. 5 – x 2. (3 – x) (4 – x) (5 – x) = 24 3. w = 2, l = 3, h = 4 Lesson 3 Let’s Try this 1 a. (w) (w – 2) (w + 4) = 8 (w + 4) b. 12 cm by 4 cm by 2 cm 2 a. (w) (w + 1) (2w + 1) = 84 b. 4 cm by 3 cm by 7 cm 3 a. (w) (w + 2) (2w + 3) = 264 b. 6 ft by 4 ft by 11 ft Lesson 3 What’s More Activity 1 1. (d) (d + 5) (d + 35) = 2000 2. (x) (12x – 15) (12x – 21) = 945 3. (x) (x – 5) (x – 5) = 250 4. (e + 6) (e + 12) (e – 4) = 2e3 5. (12 – x) (4 – x) (4 + x) = Lesson 3 What I Know 1 C 6 A 11 B 2 A 7 C 12 C 3 D 8 A 13 C 4 A 9 B 14 A 5 B 10 A 15 A Lesson 3 What’s More Activity 2 1. 40 ft by 10 ft by 5 ft 2. 21 m by 15 m by 3m 3. 10 in by 5 in by 5 in 4. e = 6cm or e = 12 cm 5. 10 in by 2 in by 6 in 44
- 55. Lesson 3 What I Have Learned 1. Read 2. Write 3. What is asked 4. Substitute 5. Length, width, height 6. Mathematical model 7. Arrange 8. Zero 9. Solve 10. Reject 11. Positive, real 12. Complete 13 – 15. 7 in by 4 in by 4 in Lesson 3 What Can I Do 1. a or 45 = a2 (a + 2) or a3 + 2a2 – 45 = 0 2. 3 ft by 3 ft by 5 ft 3. V = 32 cubic feet, the new pyramid is bigger by 17 cubic feet Lesson 3 Assessment 1 C 6 A 11 B 2 C 7 C 12 B 3 D 8 A 13 C 4 D 9 A 14 D 5 C 10 D 15 B Lesson 3 Additional Act. 3 1 – 4. h = 3w – 5, l = 2w + 3, w = w, V = 308 in3 5. (2w + 3) (w) (3w – 5) = 308 6. 3w2 – 5w 7. 6w3 – w2 – 15w – 308 = 0 8. – 308 9. (w - 4) (6w2 + 23w + 77) = 0 10 – 11. w – 4 = 0, 6w2 + 23w + 77 = 0 12. w = 4 13. w = -23/12 ± 14. 11 in by 4 in by 7 in 15. the dimension of the box is 11 in by 4 in by 7 in. Lesson 3 Additional Act. 4 1. (2h + 7) (h – 40) (h) = 0 2. V = (l – 10) (l – 10) (l) 3. (2x – 15) (5x – 21) (x) = 1935 4. (2w + 5) (w) (w – 1) = 65 5. (3w + 2) (w) (2w + 3) = 520 Assessment (Pre) 1 B 6 A 11 B 2 B 7 A 12 C 3 D 8 C 13 C 4 A 9 B 14 B 5 B 10 D 15 D Assessment (Post) 1 A 6 C 11 B 2 B 7 B 12 D 3 D 8 D 13 D 4 C 9 C 14 B 5 B 10 C 15 A 45
- 56. 46 References Callanta, Melvin M., [et.al.]. Mathematics - Grade 10 Learner’s Module. Manila City: Rex Book Store, Inc., 2015. Cabral, Emmanuela A., [et.al.]. Precalculus. Quezon City: Ateneo de Manila University Press, 2010. Oronce, Orlando A., and Marilyn O. Mendoza. E-math: Worktext in Mathematics 10. Manila City: Rex Book Store, Inc., 2015.
- 57. For inquiries and feedback, please write or call: Department of Education –Learning Resources Management and Development Center (LRMDC) DepEd Division of Bukidnon Fortich St., Sumpong, Malaybalay City, Bukidnon Telefax: ((08822)855-0048 E-mail Address: bukidnon@deped.gov.ph