Questions: Question 3 of 15 , Step 1 of 1 Correct In an effort to control vegetation overgrowh, 113 rabbits are released in an isolated area free of predators. After 2 years, it is estimated that the rabbit population has increased to 1017. Assuming exponential population growth, what will the population be after another 3 months? Round to the nearest rabbit.

Question 3 of 15 , Step 1 of 1
Correct

In an effort to control vegetation overgrowh, 113 rabbits are released in an isolated area free of predators. After 2 years, it is estimated that the rabbit population has increased to 1017. Assuming exponential population growth, what will the population be after another 3 months? Round to the nearest rabbit.

Transcript text: Question 3 of 15 , Step 1 of 1 Correct In an effort to control vegetation overgrowh, 113 rabbits are released in an isolated area free of predators. After 2 years, it is estimated that the rabbit population has increased to 1017. Assuming exponential population growth, what will the population be after another 3 months? Round to the nearest rabbit.

Solution

Solution Steps

Step 1: Calculate the Growth Rate

To find the growth rate \(r\), we use the formula \(r = \frac{1}{t} \ln\left(\frac{P_t}{P_0}\right)\). Substituting \(P_0 = 113\), \(P_t = 1017\), and \(t = 2\) into the formula gives us \(r = \frac{1}{2} \ln\left(\frac{1017}{113}\right) = 1.099\).

Step 2: Predict the Future Population

Using the growth rate \(r\) calculated, we predict the future population \(P_{future}\) after an additional time period \(t_{future} = 0.25\) years. The formula for future population is \(P_{future} = P_0 \cdot e^{r(t + t_{future})}\). Substituting the values gives us \(P_{future} = 113 \cdot e^{1.099(2 + 0.25)} = 1338\).