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The problem involves modeling the decline of a frog population using an exponential function. Let's break down the information and find the model:

1. **Initial Population (t = 0)**: Two years ago, the population was 40,000 frogs.
2. **Population after 2 years**: This year (t = 2), the population was found to be 27,000 frogs.

We want to find an exponential model of the form:

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

Where:
- \( f_0 \) = initial population (40,000).
- \( t \) = time in years.
- \( k \) = decay rate.

### Step 1: Identify Known Values
- At \( t = 0 \), \( f(0) = 40,000 \).
- At \( t = 2 \), \( f(2) = 27,000 \).

### Step 2: Find the Decay Rate (\( k \))
Using the formula \( f(t) = f_0 \cdot e^{kt} \), we can plug in \( t = 2 \) and solve for \( k \):

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

Dividing both sides by 40,000 gives:

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

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

Taking the natural logarithm on both sides:

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

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

Calculating \( k \):

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

### Step 3: Write the Exponential Model
Now that we have \( k \approx -0.196 \), the model is:

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

This function describes the decline of the frog population over time.

Would you like more details or have any questions about this solution? Here are some related questions you might find interesting:

1. How does the negative value of \( k \) influence the population model?
2. What would the population be after 5 years?
3. How long will it take for the population to decline to 10,000 frogs?
4. How would the model change if the population were increasing rather than decreasing?
5. Can this model be used for other types of animal populations? Why or why not?

**Tip**: When dealing with exponential decay models, always pay attention to the sign of the decay rate (\( k \)). A negative \( k \) indicates decay, while a positive \( k \) indicates growth.

Frogs have been dying like flies at the Enormous State University (ESU). 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, the pledge class discovered that the tags had fallen off, and they tagged a total of 27,000 frogs. Find an exponential model for the frog population, with t in years.

Discover how to model the decline of a frog population using exponential decay at Enormous State University. Two years ago, 40,000 frogs were tagged, but this year, only 27,000 were tagged. Learn how to build an exponential decay function to describe this change over time. Suitable for high school students in Grades 10-12.

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