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The problem presented in the image involves the following scenario:

- Two years ago (at \( t = 0 \)), the fraternity tagged 40,000 frogs on campus.
- This year (two years later, at \( t = 2 \)), they tagged 27,000 frogs, implying a decrease in the population.

### (a) Exponential Model for Frog Population

To find the exponential decay model, we assume the population of frogs follows the general form:

\[
f(t) = f_0 e^{-kt}
\]

where:
- \( f(t) \) is the population at time \( t \),
- \( f_0 \) is the initial population (40,000),
- \( k \) is the rate of decay,
- \( t \) is the time in years.

Using the information:
- At \( t = 0 \), \( f(0) = 40,000 \).
- At \( t = 2 \), \( f(2) = 27,000 \).

We can solve for \( k \) by substituting these values into the equation:

\[
27,000 = 40,000 e^{-2k}
\]

First, divide both sides by 40,000:

\[
\frac{27,000}{40,000} = e^{-2k}
\]

Simplifying:

\[
0.675 = e^{-2k}
\]

Now take the natural logarithm (ln) of both sides:

\[
\ln(0.675) = -2k
\]

Solving for \( k \):

\[
k = -\frac{\ln(0.675)}{2} \approx 0.196
\]

Thus, the exponential model for the frog population is:

\[
f(t) = 40,000 e^{-0.196t}
\]

### (b) Number of Tags Needed for Next Year

Next, we need to predict the population for the next year, \( t = 3 \). Using the model:

\[
f(3) = 40,000 e^{-0.196 \times 3}
\]

First, calculate the exponent:

\[
f(3) = 40,000 e^{-0.588} \approx 40,000 \times 0.555
\]

\[
f(3) \approx 22,200
\]

So, for next year's tagging, they should order approximately **22,200 tags**.

---

Would you like any further details or have any questions? Here are some follow-up questions to deepen understanding:

1. How would the exponential model change if the decay rate was different?
2. Can you calculate the frog population 5 years from now using this model?
3. What would happen to the model if more frogs were added each year due to reproduction?
4. How sensitive is the population to small changes in the decay constant \( k \)?
5. Could we predict when the frog population will drop below 10,000?

### Tip:
Always check if the context involves growth or decay before choosing your model type (exponential growth vs. decay).

Frogs have been dying like flies at the Enormous State University (ESU) campus. Each year, the pledge class of the Epsilon Delta fraternity is instructed to tag all the frogs residing on the ESU campus. Two years ago (t = 0) they managed to tag all 40,000 of them. This year's pledge class discovered that all the tags had fallen off, and they wound up tagging a total of 27,000 frogs. (a) Find an exponential model for the frog population, with t in years. (b) Assuming exponential population decrease and that all this year's tags fall off, how many tags should Epsilon Delta order for next year's pledge class?

Solve a real-world problem involving the exponential decay of the frog population at Enormous State University. Learn how to find the exponential model and predict next year's frog count using algebra and natural logarithms. Suitable for high school seniors or college students.

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