High School

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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. 3520
B. 371
C. 24
D. 225

Answer :

To find the approximate value of [tex]\( P \)[/tex], we have the function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex]. Given that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], we need to solve for [tex]\( P \)[/tex].

The equation becomes:

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

This simplifies to:

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

To find [tex]\( P \)[/tex], we solve:

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

Here, [tex]\( e^{0.25} \approx 1.284 \)[/tex].

So, substituting the approximate value:

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

[tex]\[
P \approx 225
\][/tex]

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

The correct answer choice is D. 225.

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