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 54,000 of them (with little Epsilon Delta Fraternity tags). This year's pledge class discovered that all the tags had fallen off, and they wound up tagging a total of 36,000 frogs.
(a)
Find an exponential model for the frog population, with t in years.
f(t) = 
  
Correct: Your answer is correct.
(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? (Round your answer to the nearest integer.)
 
Incorrect: Your answer is incorrect.
 tags

This problem involves finding an exponential decay model for the frog population at Enormous State University (ESU).

### Part (a) - Exponential Model
The problem states that the population of frogs two years ago was 54,000, and now it has decreased to 36,000 frogs. We need to express this population change as an exponential decay model in the form:

\[
f(t) = P_0 e^{kt}
\]

where:
- \( f(t) \) is the frog population at time \( t \),
- \( P_0 \) is the initial population (when \( t = 0 \)),
- \( k \) is the decay constant,
- \( t \) is the time in years.

Given:
- At \( t = 0 \), the initial population \( P_0 = 54,000 \),
- At \( t = 2 \) years, \( f(2) = 36,000 \).

#### Step 1: Set up the model at \( t = 0 \)
When \( t = 0 \), we know \( f(0) = P_0 = 54,000 \), so the equation is:

\[
f(0) = 54,000 = 54,000 e^{k \cdot 0} \Rightarrow f(0) = 54,000
\]

#### Step 2: Use the population at \( t = 2 \) to find \( k \)
We know that after 2 years, the population is 36,000. Substituting this into the exponential equation:

\[
36,000 = 54,000 e^{2k}
\]

Now, solve for \( k \):

\[
\frac{36,000}{54,000} = e^{2k}
\]
\[
\frac{2}{3} = e^{2k}
\]

Take the natural logarithm of both sides:

\[
\ln\left(\frac{2}{3}\right) = 2k
\]
\[
k = \frac{1}{2} \ln\left(\frac{2}{3}\right)
\]

Now, calculate \( k \):

\[
k \approx \frac{1}{2} \cdot (-0.4055) = -0.2027
\]

Thus, the exponential model for the frog population is:

\[
f(t) = 54,000 e^{-0.2027 t}
\]

### Part (b) - Frog Population Next Year
Next, we are asked to predict how many frogs will be left next year. This means we want to find \( f(t) \) when \( t = 3 \) (since it has been 2 years so far, and we want to predict for the third year).

Using the model:

\[
f(3) = 54,000 e^{-0.2027 \cdot 3}
\]

Now, calculate \( f(3) \):

\[
f(3) = 54,000 e^{-0.6081} \approx 54,000 \cdot 0.5441 \approx 29,379
\]

Thus, the fraternity should order approximately **29,379 tags** for next year's pledge class.

---

Would you like further explanation on any of the steps, or do you have any additional questions?

### Related Questions:
1. What is the half-life of the frog population, based on the exponential decay model?
2. How would the model change if the frog population grew instead of decaying?
3. If the decay constant were halved, how would the population change over time?
4. What if the initial population was 100,000 instead of 54,000? How would that affect next year's tag order?
5. Could this model apply to other species on campus, such as birds or insects?

### Tip:
In problems involving exponential growth or decay, the key is to find the rate constant \( k \) using given values at specific times.

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 54,000 of them (with little Epsilon Delta Fraternity tags). This year's pledge class discovered that all the tags had fallen off, and they wound up tagging a total of 36,000 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?

Explore how to model the exponential decay of the frog population at Enormous State University (ESU). This problem includes finding an exponential model and predicting the number of frogs next year based on population decline. Suitable for students in Grades 10-12.

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