High School

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. 3520
B. 24
C. 225
D. 371

Answer :

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

1. Identify the given information:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]

2. Substitute the given values into the function formula:
[tex]\[
288.9 = P \times e^{0.05 \times 5}
\][/tex]

3. Calculate the exponent:
[tex]\[
e^{0.05 \times 5} = e^{0.25}
\][/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. Calculate [tex]\( e^{0.25} \)[/tex]:
- Using the given accurate calculations, [tex]\( e^{0.25} \approx 1.2840254166877414 \)[/tex].

6. Plug the value of [tex]\( e^{0.25} \)[/tex] back into the equation and divide:
[tex]\[
P = \frac{288.9}{1.2840254166877414} \approx 224.99554622932885
\][/tex]

7. Round the value to the closest option:
- The approximate value of [tex]\( P \)[/tex] is [tex]\( 225 \)[/tex].

Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{225} \)[/tex], which corresponds to option C.