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. Understand the problem: We are given a function [tex]\( f(t) = P e^{rt} \)[/tex]. We know that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], and we need to find the value of [tex]\( P \)[/tex].
2. Set up the equation:
[tex]\[
f(5) = P e^{0.05 \times 5}
\][/tex]
This simplifies to:
[tex]\[
288.9 = P e^{0.25}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
We need to isolate [tex]\( P \)[/tex] in the equation:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
4. Calculate [tex]\( e^{0.25} \)[/tex]: The value of [tex]\( e^{0.25} \)[/tex] is approximately [tex]\( 1.284 \)[/tex].
5. Calculate [tex]\( P \)[/tex]:
Divide 288.9 by 1.284 to find [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{288.9}{1.284} \approx 225
\][/tex]
This calculation leads us to conclude that the approximate value of [tex]\( P \)[/tex] is 225, which corresponds to option C.
1. Understand the problem: We are given a function [tex]\( f(t) = P e^{rt} \)[/tex]. We know that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], and we need to find the value of [tex]\( P \)[/tex].
2. Set up the equation:
[tex]\[
f(5) = P e^{0.05 \times 5}
\][/tex]
This simplifies to:
[tex]\[
288.9 = P e^{0.25}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
We need to isolate [tex]\( P \)[/tex] in the equation:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
4. Calculate [tex]\( e^{0.25} \)[/tex]: The value of [tex]\( e^{0.25} \)[/tex] is approximately [tex]\( 1.284 \)[/tex].
5. Calculate [tex]\( P \)[/tex]:
Divide 288.9 by 1.284 to find [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{288.9}{1.284} \approx 225
\][/tex]
This calculation leads us to conclude that the approximate value of [tex]\( P \)[/tex] is 225, which corresponds to option C.
