Question

the population of a town can be modeled using the formula $p = 20,000e^{0.02t}$, where $t$ is the number of years after 2012 and $p$ is the towns population. which of the following equations can be used to find the number of years after 2012 that the population will double to 40,000?
$t = \frac{\ln 2}{0.02}$
$t = \frac{\log 2}{0.02}$
$t = \frac{2}{0.02e}$
$t = \frac{\ln 20,000}{0.02}$

Explanation:

Step1: Substitute \(P = 40000\) into the formula

Given \(P=20000e^{0.02t}\), when \(P = 40000\), we have \(40000=20000e^{0.02t}\).

Step2: Simplify the equation

Divide both sides by \(20000\): \(\frac{40000}{20000}=e^{0.02t}\), so \(2 = e^{0.02t}\).

Step3: Take the natural logarithm of both sides

Using the property \(\ln(e^{x})=x\), if \(2 = e^{0.02t}\), then \(\ln(2)=\ln(e^{0.02t})\). So \(\ln(2)=0.02t\).

Step4: Solve for \(t\)

Divide both sides by \(0.02\): \(t=\frac{\ln(2)}{0.02}\).

Answer:

\(t=\frac{\ln 2}{0.02}\) (the first option)