6 points
Frogs - A species of frog's population grows $24 \%$ every year. Suppose 100 frogs are released into a pond.
A. Construct an exponential model for this population.
\[
A=\text { type your answer... ( type your answer... })^{\wedge} \mathrm{t}
\]
B. How long will it take the population to reach at least 2000 frogs? type your answer... years Round to a whole number.

Step 1 ：The problem is asking for an exponential model for the frog population and the time it will take for the population to reach at least 2000 frogs. The exponential model can be represented as \(A = P*(1 + r)^t\), where \(A\) is the final amount, \(P\) is the principal amount (initial population), \(r\) is the rate of growth, and \(t\) is the time. In this case, \(P = 100\), \(r = 24\% = 0.24\).

Step 2 ：Substituting the values for \(P\) and \(r\) into the exponential model, we get \(A = 100*(1 + 0.24)^t\).

Step 3 ：To find out how long it will take for the population to reach at least 2000 frogs, we need to solve the exponential equation for \(t\) when \(A = 2000\). This will require some algebra and possibly the use of a logarithm to solve for \(t\).

Step 4 ：Solving for \(t\) when \(A = 2000\), we get \(t = 14\).

Step 5 ：The exponential model for the frog population is \(A = 100*(1 + 0.24)^t\). It will take approximately \(\boxed{14}\) years for the population to reach at least 2000 frogs.