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------------------------------------------------ If [tex]f(5) = 288.9[/tex] when [tex]r = 0.05[/tex] for the function [tex]f(t) = P e^{rt}[/tex], then what is the approximate value of [tex]P[/tex]?

A. 371
B. 3520
C. 225
D. 24

To solve this problem, we need to find the value of [tex]\( P \)[/tex] using the function [tex]\( f(t) = P e^{rt} \)[/tex]. We're given the values:

- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]

The function becomes [tex]\( f(t) = P e^{0.05 \times 5} \)[/tex].

Here is how you calculate it step-by-step:

1. Substitute the Given Values:
Plug the values into the function:
[tex]\[
288.9 = P \times e^{0.25}
\][/tex]
because [tex]\( 0.05 \times 5 = 0.25 \)[/tex].

2. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]

3. Calculate [tex]\( e^{0.25} \)[/tex]:
Using the approximate value for [tex]\( e^{0.25} \)[/tex].

4. Calculate [tex]\( P \)[/tex]:
Divide [tex]\( 288.9 \)[/tex] by the calculated value of [tex]\( e^{0.25} \)[/tex] to find [tex]\( P \)[/tex].

5. Round the Result:
Round the result to the nearest integer since the options provided are integers.

After completing these steps, we find that the approximate value of [tex]\( P \)[/tex] is 225.

Therefore, the correct answer is C. 225.