Chapter 7.7, Problem 16P

A wildlife manager determines that there are approximately 200 deer in a certain state park.

a. The population is growing at a rate of $7\%$ per year. How many deer will live in the park after 4 years?

b. If the carrying capacity of this park is 350 deer, how long will it take for the deer population to reach carrying capacity?

To determine

To calculate: A wildlife manager determines that there are approximately 200 deer in a certain state park. The population is growing at the rate of 7% per year. How many deer will live in the park after 4 years?

Answer to Problem 16P

After 4 years 262 deer will live in the park

Explanation of Solution

Given information: The number of deer is 200 and the growing rate is 7%

Calculation: Here we can use compound interest formula of exponential growth formula

Using $y=a*b^x$ where $a$ = number of deer and $b$ = rate growth factor

Here rate growth factor is $b=\left(1+\frac{7}{100}\right)=1.07$ and $x=4$ , we get

  $y=200*{1.07}^4$

  $y=200*1.31$

  $y\approx 262$ deer

Conclusion:

To determine

To calculate: A wildlife manager determines that there are approximately 200 deer in a certain state park. If the carrying capacity of this park is 350 deer, how long it will take for the deer population to reach carrying capacity?

Answer to Problem 16P

It will take between 8to 9 years to have 350 deer

Explanation of Solution

Given information: From part (i), we have $y=200*{1.07}^x$

Calculation: here $y$ is given as 350, so

  $350=200*{1.07}^x$

  $\frac{7}{4}={1.07}^x$

  $1.75={1.07}^x$

Taking logarithmic function both sides, we get

  $\log 1.75=x\log 1.07$

From logarithmic table, we have $\log 1.75=0.243$ and $\log 1.07=0.0239$

  $0.243=x*0.0239$

  $x=8.271$

Conclusion: It will take between 8to 9 years to have 350 deer.