Answer :
To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = Pe^{rt} \)[/tex] given that [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. Write the Formula for Exponential Growth:
The function is expressed as:
[tex]\[
f(t) = Pe^{rt}
\][/tex]
3. Substitute the Values into the Formula:
Plug the given values into the equation:
[tex]\[
288.9 = Pe^{0.05 \times 5}
\][/tex]
4. Calculate [tex]\( e^{rt} \)[/tex]:
First, compute the exponent:
[tex]\[
e^{0.05 \times 5} = e^{0.25}
\][/tex]
5. Solve for [tex]\( P \)[/tex]:
Rearrange the formula to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
6. Calculate the Approximate Value of [tex]\( P \)[/tex]:
Calculate the value using the approximation for [tex]\( e^{0.25} \)[/tex], and subsequently find:
[tex]\[
P \approx \frac{288.9}{1.284} \approx 225
\][/tex]
The approximate value of [tex]\( P \)[/tex] is 225. Therefore, the correct answer is option C.
1. Identify the Given Information:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]
2. Write the Formula for Exponential Growth:
The function is expressed as:
[tex]\[
f(t) = Pe^{rt}
\][/tex]
3. Substitute the Values into the Formula:
Plug the given values into the equation:
[tex]\[
288.9 = Pe^{0.05 \times 5}
\][/tex]
4. Calculate [tex]\( e^{rt} \)[/tex]:
First, compute the exponent:
[tex]\[
e^{0.05 \times 5} = e^{0.25}
\][/tex]
5. Solve for [tex]\( P \)[/tex]:
Rearrange the formula to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
6. Calculate the Approximate Value of [tex]\( P \)[/tex]:
Calculate the value using the approximation for [tex]\( e^{0.25} \)[/tex], and subsequently find:
[tex]\[
P \approx \frac{288.9}{1.284} \approx 225
\][/tex]
The approximate value of [tex]\( P \)[/tex] is 225. Therefore, the correct answer is option C.
