Answer :
To solve this problem, we need to find the value of [tex]\( P \)[/tex] using the function [tex]\( f(t) = P e^{rt} \)[/tex]. We're given the values:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]
The function becomes [tex]\( f(t) = P e^{0.05 \times 5} \)[/tex].
Here is how you calculate it step-by-step:
1. Substitute the Given Values:
Plug the values into the function:
[tex]\[
288.9 = P \times e^{0.25}
\][/tex]
because [tex]\( 0.05 \times 5 = 0.25 \)[/tex].
2. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
3. Calculate [tex]\( e^{0.25} \)[/tex]:
Using the approximate value for [tex]\( e^{0.25} \)[/tex].
4. Calculate [tex]\( P \)[/tex]:
Divide [tex]\( 288.9 \)[/tex] by the calculated value of [tex]\( e^{0.25} \)[/tex] to find [tex]\( P \)[/tex].
5. Round the Result:
Round the result to the nearest integer since the options provided are integers.
After completing these steps, we find that the approximate value of [tex]\( P \)[/tex] is 225.
Therefore, the correct answer is C. 225.
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]
The function becomes [tex]\( f(t) = P e^{0.05 \times 5} \)[/tex].
Here is how you calculate it step-by-step:
1. Substitute the Given Values:
Plug the values into the function:
[tex]\[
288.9 = P \times e^{0.25}
\][/tex]
because [tex]\( 0.05 \times 5 = 0.25 \)[/tex].
2. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
3. Calculate [tex]\( e^{0.25} \)[/tex]:
Using the approximate value for [tex]\( e^{0.25} \)[/tex].
4. Calculate [tex]\( P \)[/tex]:
Divide [tex]\( 288.9 \)[/tex] by the calculated value of [tex]\( e^{0.25} \)[/tex] to find [tex]\( P \)[/tex].
5. Round the Result:
Round the result to the nearest integer since the options provided are integers.
After completing these steps, we find that the approximate value of [tex]\( P \)[/tex] is 225.
Therefore, the correct answer is C. 225.
