X 32105 Assume that the current population of a large region

X 3.2.105 Assume that the current population of a large region is 251 million t was reported hathe population grew 9.9% since was as measured ear re ough, se the model A-Aekt. Complete parts a through c below. a. What was the population when it was measured 10 years ago? The population was 228.4 million the first time it was measured. (Round to one decimal place as needed.) b. What was the annual rate of growth between the two measurements? The annual rate of growth was 0.94% (Round to two decimal places as needed.) c. Assuming the same rate of growth, determine in how many years from the current year the region will have a population of 400 million. The region will have a population of 400 million in 50 years Round up to the nearest year.)

Solution

Let the population before 10 years be x

According to question

(251 - x )/ x =9.9%

=> 251/x = 9.9/100 +1

=> 251/x = 109.9/100

=.> x = (251*100) / 109.9

=> x = 229.01 million

Find k constant

We know A = 109.9/100 (A_o)

using eqn A= A_o e^kt

=> 109.9/100(A_o) =A_o e^10k

taking log both side

=> ln(1.099) = 10k ln(e)

=> k = 0.0944006/10

=> k = 0.00944006

b) annual growth rate = 9.9%/10 = 0.99%

c) A= 400 million

A_o= 251 million

k = 0.00944006

=> 400 = 251 e^(0.00944006t)

=> 400/251 = e^(0.00944006t)

taking log both sides

=> ln(1.593625) =0.00944006t ln(e)

=> t = 0.466011608/0.00944006

=> t = 49.3653 ~ 49 years