Worked solutions for example 1 Workshop 2 - Basic Algebra OES Contextual problems

1. 1a Write an equation that can be used to work out the
volume of water entering the basin without knowing the
volume of water leaving the basin.

2. 1a • Substitute V2 in the second equation with its equivalent from the first
equation:

3. 1a • Substitute V2 in the second equation with its equivalent from the first
equation:
V1S1 = (V1 + W)S2

4. 1a • Substitute V2 in the second equation with its equivalent from the first
equation:
V1S1 = (V1 + W)S2
• Then rearrange for V1:

5. 1a • Substitute V2 in the second equation with its equivalent from the first
equation:
V1S1 = (V1 + W)S2
• Then rearrange for V1:
V1S1 = V1S2 + WS2

6. 1a • Substitute V2 in the second equation with its equivalent from the first
equation:
V1S1 = (V1 + W)S2
• Then rearrange for V1:
V1S1 = V1S2 + WS2
V1S1 − V1S2 = WS2

7. 1a • Substitute V2 in the second equation with its equivalent from the first
equation:
V1S1 = (V1 + W)S2
• Then rearrange for V1:
V1S1 = V1S2 + WS2
V1S1 − V1S2 = WS2
V1 S1 − S2 = WS2

8. 1a • Substitute V2 in the second equation with its equivalent from the first
equation:
V1S1 = (V1 + W)S2
• Then rearrange for V1:
V1S1 = V1S2 + WS2
V1S1 − V1S2 = WS2
V1 S1 − S2 = WS2
퐕ퟏ =
퐖퐒ퟐ
퐒ퟏ − 퐒ퟐ

9. 1b It is important to be able to calculate the additions to the
water also without knowing the volume of water leaving
the basin. Write an equation to do this.

10. 1b • Rearrange your previous equation to make W the subject:

11. 1b • Rearrange your previous equation to make W the subject:
V1 =
WS2
S1 − S2

12. 1b • Rearrange your previous equation to make W the subject:
V1 =
WS2
S1 − S2
V1 S1 − S2 = WS2

13. 1b • Rearrange your previous equation to make W the subject:
V1 =
WS2
S1 − S2
V1 S1 − S2 = WS2
퐕ퟏ 퐒ퟏ − 퐒ퟐ
퐒ퟐ
= 퐖

14. 1c Over a year, a lagoon of surface area 50m2 gains 1.12m
of water by precipitation and 0.81m by run-off, but loses
1.26m by evaporation. What is W for the lake in m3s-1?

15. 1c • To convert m/year into m3s-1, you need to multiply it by the area if the
basin and divide by the number of seconds in a year:

16. 1c • To convert m/year into m3s-1, you need to multiply it by the area if the
basin and divide by the number of seconds in a year:
1 year = 356 x 24 x 60 x 60 seconds = 31536000s

17. 1c • To convert m/year into m3s-1, you need to multiply it by the area if the
basin and divide by the number of seconds in a year:
1 year = 356 x 24 x 60 x 60 seconds = 31536000s
P = 1.12my−1 =
1.12 x 50
31536000
= 1.78 x 10−6m3s−1

18. 1c • To convert m/year into m3s-1, you need to multiply it by the area if the
basin and divide by the number of seconds in a year:
1 year = 356 x 24 x 60 x 60 seconds = 31536000s
P = 1.12my−1 =
1.12 x 50
31536000
= 1.78 x 10−6m3s−1
R = 0.81my−1 =
0.81 x 50
31536000
= 1.28 x 10−6m3s−1

19. 1c • To convert m/year into m3s-1, you need to multiply it by the area if the
basin and divide by the number of seconds in a year:
1 year = 356 x 24 x 60 x 60 seconds = 31536000s
P = 1.12my−1 =
1.12 x 50
31536000
= 1.78 x 10−6m3s−1
R = 0.81my−1 =
0.81 x 50
31536000
= 1.28 x 10−6m3s−1
E = 1.26my−1 =
1.26 x 50
31536000
= 2.00 x 10−6m3s−1

20. 1c • To convert m/year into m3s-1, you need to multiply it by the area if the
basin and divide by the number of seconds in a year:
1 year = 356 x 24 x 60 x 60 seconds = 31536000s
P = 1.12my−1 =
1.12 x 50
31536000
= 1.78 x 10−6m3s−1
R = 0.81my−1 =
0.81 x 50
31536000
= 1.28 x 10−6m3s−1
E = 1.26my−1 =
1.26 x 50
31536000
= 2.00 x 10−6m3s−1
• Put the converted P, R and E back into the equation for W:

21. 1c • To convert m/year into m3s-1, you need to multiply it by the area if the
basin and divide by the number of seconds in a year:
1 year = 356 x 24 x 60 x 60 seconds = 31536000s
P = 1.12my−1 =
1.12 x 50
31536000
= 1.78 x 10−6m3s−1
R = 0.81my−1 =
0.81 x 50
31536000
= 1.28 x 10−6m3s−1
E = 1.26my−1 =
1.26 x 50
31536000
= 2.00 x 10−6m3s−1
• Put the converted P, R and E back into the equation for W:
W = 1.78 + 1.28 − 2 x 106m3s−1 = ퟏ. ퟎퟔ 퐱 ퟏퟎ−ퟔ퐦ퟑ퐬−ퟏ

22. 1d Seawater of salinity 35 seeps into the lagoon at a rate of
5.4 x 10-5m3s-1 which has a salinity of exactly 35. What
will the salinity of the lagoon be after at least a year?

23. 1d • Substitute V2 in the second equation with its equivalent from
the first equation:

24. 1d • Substitute V2 in the second equation with its equivalent from
the first equation:
V1S1 = (V1 + W)S2

25. 1d • Substitute V2 in the second equation with its equivalent from
the first equation:
V1S1 = (V1 + W)S2
• Rearrange for S2 and substitute in the values you know:

26. 1d • Substitute V2 in the second equation with its equivalent from
the first equation:
V1S1 = (V1 + W)S2
• Rearrange for S2 and substitute in the values you know:
V1S1
(V1 + W)
= S2

27. 1d • Substitute V2 in the second equation with its equivalent from
the first equation:
V1S1 = (V1 + W)S2
• Rearrange for S2 and substitute in the values you know:
V1S1
(V1 + W)
= S2
S2 =
5.4 x 10−5 x 35
5.4 x 10−5 + 1.06 x 10−6 = ퟑퟒ. ퟑퟑ