Question

The population of a town grew from 20,000 to 28,000. The continuous growth rate is 15%. The equation 20,000e^0.15t=28,000 represents the situation, where t is the number of years the population has been growing. About how many years has the population of the town been growing? Use a calculator and round your answer to the nearest whole number.

A. 2 years
B.9 years
C. 17 years
D. 22 years

Answer 1

The correct option is:  A.  2 years

Explanation

The given growth equation is:   $20000e^0^.^1^5^t = 28000$, where  $t$ is the number of years the population has been growing.

For finding the number of years, we will solve the above equation for  $t$.

First, dividing both sides by 20000, we will get........

$\frac{20000e^0^.^1^5^t}{20000}=\frac{28000}{20000}\\ \\ e^0^.^1^5^t = 1.4$

Now taking 'natural log' on both sides, we will get........

$ln(e^0^.^1^5^t)=ln(1.4)\\ \\ 0.15t*ln(e)= ln(1.4)\\ \\ 0.15t*1=ln(1.4)\\ \\ t=\frac{ln(1.4)}{0.15}=2.243..... \approx 2$

So, the population of the town has been growing about 2 years.

Answer 2

The town has been growing for 17 years

Exponential functions

Given the exponential function

Given the expressions 20,000e^0.15t=28,000

We are to find the value of "t" which is the time.

e^0.15t = 28000/20000

e^0.15t = 7/5

e^0.15t = 1.4

Take the ln of both sides

lne^0.15t = ln 1.4

0.15t = 2.639

t = 2.639/0.15

t = 17years

Hence the town has been growing for 17 years

Learn more on exponential functions here: https://brainly.com/question/2456547