Question

An animal reserve has 44,000 elk. The population is increasing at a rate of 15% per year. How long will ittake for the population to reach 88,000? If necessary, round your answer to the nearest tenth.The population will reach 88,000 in approximately _____years.

Answer 1

An animal reserve has 44,000 elk. The population is increasing at a rate of 15% per year. How long will it

take for the population to reach 88,000? If necessary, round your answer to the nearest tenth.

The population will reach 88,000 in approximately _____

years.

In this problem we have an exponential function of the form

$y=a(1+r)^x$

where

a is the initial value

r is the rate

y ----> is the population of elk

x -----> is the number of years

we have

a=44,000

r=15%=15/100=0.15

substitute

$\begin{gathered} y=44,000(1+0.15)^x \\ y=44,000(1.15)^x \end{gathered}$

For y=88,000

substitute in the equation

$\begin{gathered} 88,000=44,000(1.15)^x \\ \frac{88,000}{44,000}=1.15^x \\ \\ 2=1.15^x \\ \text{apply log both sides} \\ \log (2)=x\cdot\log (1.15) \\ x=4.96\text{ years} \end{gathered}$

therefore

The population will reach 88,000 in approximately 5 years