Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex] when [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], follow these steps:
1. Start with the given function:
[tex]\( f(t) = P e^{rt} \)[/tex]
2. Substitute the known values:
We know that [tex]\( f(5) = 288.9 \)[/tex], [tex]\( r = 0.05 \)[/tex], and [tex]\( t = 5 \)[/tex]. So, the equation becomes:
[tex]\[
288.9 = P e^{0.05 \times 5}
\][/tex]
3. Simplify the exponent:
Calculate the exponent:
[tex]\[
0.05 \times 5 = 0.25
\][/tex]
Therefore, the equation becomes:
[tex]\[
288.9 = P e^{0.25}
\][/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 a calculator, we find:
[tex]\[
e^{0.25} \approx 1.284025
\][/tex]
6. Calculate [tex]\( P \)[/tex]:
Plug the value of [tex]\( e^{0.25} \)[/tex] into the equation:
[tex]\[
P = \frac{288.9}{1.284025} \approx 224.995
\][/tex]
7. Choose the closest value:
Compare the calculated value of [tex]\( P \approx 224.995 \)[/tex] with the given options:
- A. 371
- B. 24
- C. 225
- D. 3520
The closest value to 224.995 is [tex]\( C. 225 \)[/tex].
Thus, the approximate value of [tex]\( P \)[/tex] is [tex]\(\boxed{225}\)[/tex].
1. Start with the given function:
[tex]\( f(t) = P e^{rt} \)[/tex]
2. Substitute the known values:
We know that [tex]\( f(5) = 288.9 \)[/tex], [tex]\( r = 0.05 \)[/tex], and [tex]\( t = 5 \)[/tex]. So, the equation becomes:
[tex]\[
288.9 = P e^{0.05 \times 5}
\][/tex]
3. Simplify the exponent:
Calculate the exponent:
[tex]\[
0.05 \times 5 = 0.25
\][/tex]
Therefore, the equation becomes:
[tex]\[
288.9 = P e^{0.25}
\][/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 a calculator, we find:
[tex]\[
e^{0.25} \approx 1.284025
\][/tex]
6. Calculate [tex]\( P \)[/tex]:
Plug the value of [tex]\( e^{0.25} \)[/tex] into the equation:
[tex]\[
P = \frac{288.9}{1.284025} \approx 224.995
\][/tex]
7. Choose the closest value:
Compare the calculated value of [tex]\( P \approx 224.995 \)[/tex] with the given options:
- A. 371
- B. 24
- C. 225
- D. 3520
The closest value to 224.995 is [tex]\( C. 225 \)[/tex].
Thus, the approximate value of [tex]\( P \)[/tex] is [tex]\(\boxed{225}\)[/tex].
