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

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

Answer :

Sure, let's solve the problem step-by-step.

We are given a function [tex]\( f(t) = P e^{rt} \)[/tex] and specific values:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]

We need to find the approximate value of [tex]\( P \)[/tex].

### Step-by-Step Solution

1. Substitute the known values into the function:

[tex]\( f(5) = P e^{0.05 \cdot 5} \)[/tex]

2. We know [tex]\( f(5) = 288.9 \)[/tex], so the equation becomes:

[tex]\( 288.9 = P e^{0.25} \)[/tex]

3. Calculate [tex]\( e^{0.25} \)[/tex]:

The value of [tex]\( e^{0.25} \)[/tex] is approximately 1.284.

Thus, the equation is:

[tex]\( 288.9 = P \cdot 1.284 \)[/tex]

4. Solve for [tex]\( P \)[/tex]:

Divide both sides by [tex]\( 1.284 \)[/tex]:

[tex]\( P = \frac{288.9}{1.284} \)[/tex]

[tex]\( P \approx 224.995546229 \)[/tex]

5. Identify the closest value from the given options:

The given options are:
- A. 371
- B. 3520
- C. 24
- D. 225

The value we calculated, approximately 225, is closest to option D.

### Answer:
The approximate value of [tex]\( P \)[/tex] is 225.
Option D is the correct answer.