Question

1. the beaver population on kodiak island has an annual growth rate of 1.2% per year. approximate the number of years it will take for the population to double. if the initial population is 32 beavers, use it to find the population of beavers on the island in 20 years.

Explanation:

Step1: Use the continuous - growth formula \(P = P_0e^{rt}\)

Let \(P_0\) be the initial population, \(r\) be the growth rate, and \(t\) be the time. When the population doubles, \(P = 2P_0\). Given \(r=0.012\), the equation becomes \(2P_0=P_0e^{0.012t}\). Divide both sides by \(P_0\) (since \(P_0
eq0\)), we get \(e^{0.012t}=2\).

Step2: Solve for \(t\) using the natural logarithm

Take the natural logarithm of both sides: \(\ln(e^{0.012t})=\ln(2)\). By the property \(\ln(e^{x}) = x\), we have \(0.012t=\ln(2)\). Then \(t=\frac{\ln(2)}{0.012}\approx\frac{0.6931}{0.012}\approx57.76\) years.

Step3: Find the population in 20 years

Given \(P_0 = 32\), \(r = 0.012\), and \(t = 20\). Use the formula \(P=P_0e^{rt}\). Substitute the values: \(P = 32e^{0.012\times20}=32e^{0.24}\). Since \(e^{0.24}\approx1.2712\), then \(P\approx32\times1.2712 = 40.68\approx41\)

Answer:

It will take approximately \(58\) years for the population to double. The population of beavers on the island in \(20\) years is approximately \(41\).