Answer :
To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex], when it's given that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], we can follow these steps:
1. Substitute the known values into the equation:
[tex]\[
288.9 = P \cdot e^{0.05 \times 5}
\][/tex]
2. Calculate the exponent [tex]\( r \times t \)[/tex]:
[tex]\[
r \times t = 0.05 \times 5 = 0.25
\][/tex]
3. Evaluate [tex]\( e^{0.25} \)[/tex]:
The approximate value of [tex]\( e^{0.25} \)[/tex] is 1.2840 (rounded to four decimal places).
4. Solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{1.2840}
\][/tex]
[tex]\[
P \approx 224.995
\][/tex]
From these calculations, [tex]\( P \)[/tex] is approximately 225.
Therefore, the correct answer is:
B. 225
1. Substitute the known values into the equation:
[tex]\[
288.9 = P \cdot e^{0.05 \times 5}
\][/tex]
2. Calculate the exponent [tex]\( r \times t \)[/tex]:
[tex]\[
r \times t = 0.05 \times 5 = 0.25
\][/tex]
3. Evaluate [tex]\( e^{0.25} \)[/tex]:
The approximate value of [tex]\( e^{0.25} \)[/tex] is 1.2840 (rounded to four decimal places).
4. Solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{1.2840}
\][/tex]
[tex]\[
P \approx 224.995
\][/tex]
From these calculations, [tex]\( P \)[/tex] is approximately 225.
Therefore, the correct answer is:
B. 225
