Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex], with the given information that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], follow these steps:
1. Substitute the Known Values:
Plug the values into the equation. We have [tex]\( f(5) = 288.9 \)[/tex], [tex]\( t = 5 \)[/tex], and [tex]\( r = 0.05 \)[/tex]. The equation becomes:
[tex]\[
288.9 = P \times e^{0.05 \times 5}
\][/tex]
2. Calculate the Exponent:
First, calculate the exponent [tex]\( 0.05 \times 5 \)[/tex], which equals [tex]\( 0.25 \)[/tex]. So the equation is:
[tex]\[
288.9 = P \times e^{0.25}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
To find [tex]\( P \)[/tex], we need to divide both sides of the equation by [tex]\( e^{0.25} \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
4. Approximate [tex]\( e^{0.25} \)[/tex]:
Using a calculator, approximate [tex]\( e^{0.25} \)[/tex].
5. Calculate [tex]\( P \)[/tex]:
Divide 288.9 by the approximate value of [tex]\( e^{0.25} \)[/tex].
The result for [tex]\( P \)[/tex] is approximately 225.
Therefore, the correct choice is:
A. 225
1. Substitute the Known Values:
Plug the values into the equation. We have [tex]\( f(5) = 288.9 \)[/tex], [tex]\( t = 5 \)[/tex], and [tex]\( r = 0.05 \)[/tex]. The equation becomes:
[tex]\[
288.9 = P \times e^{0.05 \times 5}
\][/tex]
2. Calculate the Exponent:
First, calculate the exponent [tex]\( 0.05 \times 5 \)[/tex], which equals [tex]\( 0.25 \)[/tex]. So the equation is:
[tex]\[
288.9 = P \times e^{0.25}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
To find [tex]\( P \)[/tex], we need to divide both sides of the equation by [tex]\( e^{0.25} \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
4. Approximate [tex]\( e^{0.25} \)[/tex]:
Using a calculator, approximate [tex]\( e^{0.25} \)[/tex].
5. Calculate [tex]\( P \)[/tex]:
Divide 288.9 by the approximate value of [tex]\( e^{0.25} \)[/tex].
The result for [tex]\( P \)[/tex] is approximately 225.
Therefore, the correct choice is:
A. 225
