EXAMPLE PROBLEMS ON EXPONENTIAL GROWTH FUNCTIONS

Example 1 :

The population size of rabbits of an farm is given, approximately by R  =  50x(1.07)ⁿ where n is the number of weeks after the rabbit farm was established.

(a)  What was the original rabbit population ?

(b)  How many rabbits were present after 15 weeks ?

(c)  How many rabbits were present after 30 weeks ?

(d)  Sketch the graph of R against n (n ≥ 0)

(e) How long would it take for the population to reach 500 ?

Solution :

Given function :

R  =  50x(1.07)ⁿ

R = Population size and n = number of weeks farms was established.

(a)  To find original population, we apply n = 0

R  =  50x(1.07)⁰

R  =  50

So, there were 50 rabbits originally in the farm.

(b)  To find number of rabbits after 15 weeks, we apply n = 15

R  =  50x(1.07)¹⁵

R  =  50 x (2.76)

R  =  138 rabbits.

(c)  To find number of rabbits after 30 weeks, we apply n = 30

R  =  50x(1.07)³⁰

R  =  50 x (7.61)

R  =  381 rabbits.

(d) 

(e)  From the graph, by observing for what value of n we get the value of R as 500.

So, the answer is approximately 34 weeks.

Example 2 :

The population of nest and ants, n weeks after it is established is given by 

P  =  500 x (1.12)ⁿ

(a)  How many ants were originally in the nest ?

(b)  How many ants were in the nest after :

(i)  10  weeks  (ii)  20 weeks ?

(c)  Find how many weeks it takes for the ant population to reach 2000.

Solution :

P  =  500 x (1.12)ⁿ

(a)  Number of ants were originally (n)  =  0

P  =  500 x (1.12)⁰

P  =  500

(b)  (i)  10  weeks

n  =  10

P  =  500 x (1.12)¹⁰

P  =  500x3.11

P  =  1555

(ii)  20 weeks

P  =  500 x (1.12)²⁰

P  =  500x9.65

P  =  4825

From the graph, it will take approximately 13 weeks to reach the population of 2000. 

Example 3 :

The weight of bacteria in the culture, t hours after it has been established, is given by the formula

W(t)  =  20 x (1.007)^t grams

(a)  Find the original weight of bacteria in the culture.

(b)  The weight of the bacteria in 24 hours.

(c)  find how long it takes for the weight to reach 100 grams.

Solution :

(a)  To find original weight of bacteria, we apply t  =  0

W(0)  =  20 x (1.007)⁰ grams

W(0)  =  20 grams

(b)  At t = 24

W(24)  =  20 x (1.007)²⁴ grams

W(24)  =  23.64 grams

Approximately we will have 24 grams of bacteria in 24 hours.

Example 4 :

The population of wasps in a nest, n days after it is discovered, is given by 

P  =  250x(1.06)ⁿ

(a)  How many wasps were in the nest originally.

(b)  Find the number of wasps after   

(i)  25 days  (ii)  8 weeks

Solution :

Given  :

P  =  250x(1.06)ⁿ

(a)  Original number of wasps

n = 0

P  =  250x(1.06)⁰

P  =  250 wasps

(b)  Find the number of wasps after   

(i)  25 days 

P  =  250x(1.06)²⁵

P  =  1072.96

P  =  1073

So, approximately 1073 wasps are there in the nest.

(ii)  8 weeks

1 week  =  7 days

8 weeks  =  56 days

P  =  250x(1.06)⁵⁶

P  =  6532

After 8 weeks, there will be 6532 wasps.

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