Extension 1

1. Laws of Growth and
Decay

2. Natural Growth and
Decay
dP
= kP → P = P0 e !
kt
dt

3. Example 1
• The annual growth rate of a town is 3% of
its population. In 2000 the population is
13000.
• Show that the population can be
calculated by the general growth and
decay model
• Hence find the population in 2010 (to
nearest 100)

4. Example 2 - Half Life
The half life a a radio-active substance is 200
years. Initially there are 200g. How much
mass remains after 300 years.

5. Law of Limited Growth
and Decay
In reality the population growth may be
restricted due to death, diseases or
limitations in food, water etc
DP
If = k(P − A) → P = A + Be !
kt
dt

6. Example 3 - Natural
Growth and Decay

7. Newton’s Law of
Cooling
the cooling rate of a body is
proportional to the difference
between the temperature of the
body (T) and that of the surrounding
medium (M)
DT
= −k(T − M ) !
dt

8. Example - Newtons
Law of Cooling

9. Wilhelmy’s Law
The velocity of a reaction is proportional
to difference between the initial
concentration of the reagent (a) and the
amount transformed at any time (x)
dx
= k(a − x) 0 ≤ x ≤ a !
dt

10. Wilhelmy’s Law
Example
In a particular reaction the
concentration of a reagent is
10 (g) and 4 (g) have already
been transformed. Find the
amount left after 5 minutes

11. General Example
Brine containing 2 g of salt per litre flows
into a tank initially filled with 50 litres of
water containing 10 g of salt. If the brine
enters the tank at 5 litres /minute, the
concentration is kept uniform by stirring, and
the mixture flows out at the same rate, find
the amount of salt in the tank after 10
minutes.