Answer :
To solve this problem, you're given a function [tex]\( f(t) = P e^{rt} \)[/tex] and specific values to find the approximate value of [tex]\( P \)[/tex].
Here's how you can approach the problem step-by-step:
1. Understand the given information:
- The function is [tex]\( f(t) = P e^{rt} \)[/tex].
- You know that when [tex]\( t = 5 \)[/tex], [tex]\( f(5) = 288.9 \)[/tex].
- It is also given that [tex]\( r = 0.05 \)[/tex].
2. Rewrite the equation for the given scenario:
- Since [tex]\( f(5) = P e^{0.05 \times 5} \)[/tex], you have the equation:
[tex]\[
288.9 = P e^{0.25}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
- To find [tex]\( P \)[/tex], you need to rearrange the equation:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
4. Approximate [tex]\( e^{0.25} \)[/tex]:
- The value of [tex]\( e^{0.25} \)[/tex] is a constant. When you calculate it, it is approximately [tex]\( 1.284 \)[/tex].
5. Calculate [tex]\( P \)[/tex]:
- Substitute the approximate value of [tex]\( e^{0.25} \)[/tex]:
[tex]\[
P \approx \frac{288.9}{1.284} \approx 225
\][/tex]
Based on these calculations, the approximate value of [tex]\( P \)[/tex] is closest to 225, corresponding to option D.
Here's how you can approach the problem step-by-step:
1. Understand the given information:
- The function is [tex]\( f(t) = P e^{rt} \)[/tex].
- You know that when [tex]\( t = 5 \)[/tex], [tex]\( f(5) = 288.9 \)[/tex].
- It is also given that [tex]\( r = 0.05 \)[/tex].
2. Rewrite the equation for the given scenario:
- Since [tex]\( f(5) = P e^{0.05 \times 5} \)[/tex], you have the equation:
[tex]\[
288.9 = P e^{0.25}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
- To find [tex]\( P \)[/tex], you need to rearrange the equation:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
4. Approximate [tex]\( e^{0.25} \)[/tex]:
- The value of [tex]\( e^{0.25} \)[/tex] is a constant. When you calculate it, it is approximately [tex]\( 1.284 \)[/tex].
5. Calculate [tex]\( P \)[/tex]:
- Substitute the approximate value of [tex]\( e^{0.25} \)[/tex]:
[tex]\[
P \approx \frac{288.9}{1.284} \approx 225
\][/tex]
Based on these calculations, the approximate value of [tex]\( P \)[/tex] is closest to 225, corresponding to option D.
