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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. 24
B. 371
C. 225
D. 3520

Answer :

To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{t} \)[/tex] given that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], we can follow these steps:

1. Substitute the given values into the function:

You are provided with the function [tex]\( f(t) = P e^{t} \)[/tex]. We know that [tex]\( f(5) = 288.9 \)[/tex]. This means:

[tex]\[
288.9 = P e^{0.05 \times 5}
\][/tex]

2. Calculate the exponent:

Compute the exponent by multiplying the given rate [tex]\( r \)[/tex] by the time [tex]\( t \)[/tex]:

[tex]\[
0.05 \times 5 = 0.25
\][/tex]

3. Compute the value of [tex]\( e^{0.25} \)[/tex]:

The mathematical constant [tex]\( e \)[/tex] raised to the power of 0.25 is:

[tex]\[
e^{0.25} \approx 1.284
\][/tex]

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

Rearrange the equation to solve for [tex]\( P \)[/tex]:

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

Substituting the value of [tex]\( e^{0.25} \)[/tex], you get:

[tex]\[
P = \frac{288.9}{1.284} \approx 225
\][/tex]

Therefore, the approximate value of [tex]\( P \)[/tex] is 225. The correct answer is:

C. 225