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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. 225
B. 24
C. 3520
D. 371

Answer :

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

1. Identify Given Information:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]

2. Set Up the Equation:
The function is given by:
[tex]\[
f(t) = P e^{r \cdot t}
\][/tex]
Substituting the given values into the equation, we have:
[tex]\[
288.9 = P e^{0.05 \cdot 5}
\][/tex]

3. Calculate the Exponential Factor:
Calculate the value of the exponent:
[tex]\[
e^{0.05 \cdot 5} = e^{0.25}
\][/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]

5. Approximate Result:
Upon calculating, we find that the value of [tex]\( P \)[/tex] is approximately [tex]\( 225 \)[/tex].

6. Choose the Closest Answer:
Comparing the approximate result to the provided options, the closest answer is:
[tex]\[
\text{A. 225}
\][/tex]

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

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