Answer :
To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex] given that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], we can follow these steps:
1. Understand the function:
The function given is [tex]\( f(t) = P e^{rt} \)[/tex]. For our specific problem, we know:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]
2. Set up the equation:
We substitute the known values into the equation:
[tex]\[
288.9 = P \times e^{0.05 \times 5}
\][/tex]
Simplify the exponent:
[tex]\[
e^{0.05 \times 5} = e^{0.25}
\][/tex]
3. Calculate [tex]\( e^{0.25} \)[/tex]:
The approximate value of [tex]\( e^{0.25} \)[/tex] is around 1.284.
4. Solve for [tex]\( P \)[/tex]:
Use the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}} \approx \frac{288.9}{1.284}
\][/tex]
[tex]\[
P \approx 225
\][/tex]
5. Choose the closest answer:
By solving the above, we find [tex]\( P \)[/tex] is approximately 225. Therefore, the correct answer is:
- C. 225
This is how you determine the approximate value of [tex]\( P \)[/tex] using the given function and parameters.
1. Understand the function:
The function given is [tex]\( f(t) = P e^{rt} \)[/tex]. For our specific problem, we know:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]
2. Set up the equation:
We substitute the known values into the equation:
[tex]\[
288.9 = P \times e^{0.05 \times 5}
\][/tex]
Simplify the exponent:
[tex]\[
e^{0.05 \times 5} = e^{0.25}
\][/tex]
3. Calculate [tex]\( e^{0.25} \)[/tex]:
The approximate value of [tex]\( e^{0.25} \)[/tex] is around 1.284.
4. Solve for [tex]\( P \)[/tex]:
Use the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}} \approx \frac{288.9}{1.284}
\][/tex]
[tex]\[
P \approx 225
\][/tex]
5. Choose the closest answer:
By solving the above, we find [tex]\( P \)[/tex] is approximately 225. Therefore, the correct answer is:
- C. 225
This is how you determine the approximate value of [tex]\( P \)[/tex] using the given function and parameters.