Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex], we are given that [tex]\( f(5) = 288.9 \)[/tex] and the rate [tex]\( r = 0.05 \)[/tex].
Here's how you can find [tex]\( P \)[/tex]:
1. Start with the function:
The function is given by [tex]\( f(t) = P e^{rt} \)[/tex].
2. Plug in the given values:
We know that [tex]\( f(5) = 288.9 \)[/tex], so:
[tex]\[
288.9 = P e^{0.05 \times 5}
\][/tex]
3. Calculate [tex]\( e^{0.05 \times 5} \)[/tex]:
The expression [tex]\( 0.05 \times 5 \)[/tex] simplifies to [tex]\( 0.25 \)[/tex].
Calculate [tex]\( e^{0.25} \)[/tex].
4. Rearrange the equation to solve for [tex]\( P \)[/tex]:
Now, rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
5. Compute the value of [tex]\( P \)[/tex]:
Divide 288.9 by the value of [tex]\( e^{0.25} \)[/tex].
6. Determine the approximate value:
Upon calculation, the approximate value of [tex]\( P \)[/tex] is very close to 225.
Therefore, the closest answer choice for the value of [tex]\( P \)[/tex] is [tex]\( D. 225 \)[/tex].
Here's how you can find [tex]\( P \)[/tex]:
1. Start with the function:
The function is given by [tex]\( f(t) = P e^{rt} \)[/tex].
2. Plug in the given values:
We know that [tex]\( f(5) = 288.9 \)[/tex], so:
[tex]\[
288.9 = P e^{0.05 \times 5}
\][/tex]
3. Calculate [tex]\( e^{0.05 \times 5} \)[/tex]:
The expression [tex]\( 0.05 \times 5 \)[/tex] simplifies to [tex]\( 0.25 \)[/tex].
Calculate [tex]\( e^{0.25} \)[/tex].
4. Rearrange the equation to solve for [tex]\( P \)[/tex]:
Now, rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
5. Compute the value of [tex]\( P \)[/tex]:
Divide 288.9 by the value of [tex]\( e^{0.25} \)[/tex].
6. Determine the approximate value:
Upon calculation, the approximate value of [tex]\( P \)[/tex] is very close to 225.
Therefore, the closest answer choice for the value of [tex]\( P \)[/tex] is [tex]\( D. 225 \)[/tex].
