Question

the population of a town can be modeled using hte formula p=20,000 e^0.02t , where t is the number of years after 2012 and p is the town's population . which of the following equations can be used to find the number of years after 2012 hat the population will doubke to 40,000 ?

Answer 1

To find the number of years after 2012 that the town's population will double to 40,000, we can set up the equation based on the population model provided:

Given:

\[
p = 20,000 e^{0.02t}
\]

We want to find \( t \) when \( p = 40,000 \). So, we set up the equation:

\[
40,000 = 20,000 e^{0.02t}
\]

Now, divide both sides by 20,000:

\[
\frac{40,000}{20,000} = e^{0.02t}
\]

This simplifies to:

\[
2 = e^{0.02t}
\]

To solve for \( t \), we take the natural logarithm of both sides:

\[
\ln(2) = \ln(e^{0.02t})
\]

Using the property of logarithms that \( \ln(e^x) = x \), we have:

\[
\ln(2) = 0.02t
\]

Now we can solve for \( t \):

\[
t = \frac{\ln(2)}{0.02}
\]

This equation can be used to find the number of years after 2012 that the population will double to 40,000.