Question

in 2012, the population of a city was 5.78 million. the exponential growth rate was 2.52% p
a) find the exponential growth function.
b) estimate the population of the city in 2018.
c) when will the population of the city be 9 million?
d) find the doubling time.

a) the exponential growth function is ( p(t)=5.78e^{0.0252t} ), where ( t ) is in terms of the number
(type exponential notation with positive exponents. do not simplify. use integers or decimal
b) the population of the city in 2018 is 6.7 million.
(round to one decimal place as needed.)
c) the population of the city will be 9 million in about 17.6 years after 2012.
(round to one decimal place as needed.)
d) the doubling time is about ( square ) years.
(simplify your answer. round to one decimal place as needed.)

Explanation:

Step1: Recall the formula for doubling time

The formula for doubling time \(T\) in an exponential growth model \(P(t)=P_0e^{kt}\) is \(T = \frac{\ln 2}{k}\). Here, \(k = 0.0252\) (since the growth rate is \(2.52\%=0.0252\)).

Step2: Calculate the doubling time

Substitute \(k = 0.0252\) into the formula \(T=\frac{\ln 2}{k}\).
We know that \(\ln 2\approx0.6931\), so \(T=\frac{0.6931}{0.0252}\).

$$T=\frac{0.6931}{0.0252}\approx27.5$$

Answer:

The doubling time is about \(27.5\) years.