Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{t} \)[/tex] given that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], we can follow these steps:
1. Substitute the given values into the function:
You are provided with the function [tex]\( f(t) = P e^{t} \)[/tex]. We know that [tex]\( f(5) = 288.9 \)[/tex]. This means:
[tex]\[
288.9 = P e^{0.05 \times 5}
\][/tex]
2. Calculate the exponent:
Compute the exponent by multiplying the given rate [tex]\( r \)[/tex] by the time [tex]\( t \)[/tex]:
[tex]\[
0.05 \times 5 = 0.25
\][/tex]
3. Compute the value of [tex]\( e^{0.25} \)[/tex]:
The mathematical constant [tex]\( e \)[/tex] raised to the power of 0.25 is:
[tex]\[
e^{0.25} \approx 1.284
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
Substituting the value of [tex]\( e^{0.25} \)[/tex], you get:
[tex]\[
P = \frac{288.9}{1.284} \approx 225
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 225. The correct answer is:
C. 225
1. Substitute the given values into the function:
You are provided with the function [tex]\( f(t) = P e^{t} \)[/tex]. We know that [tex]\( f(5) = 288.9 \)[/tex]. This means:
[tex]\[
288.9 = P e^{0.05 \times 5}
\][/tex]
2. Calculate the exponent:
Compute the exponent by multiplying the given rate [tex]\( r \)[/tex] by the time [tex]\( t \)[/tex]:
[tex]\[
0.05 \times 5 = 0.25
\][/tex]
3. Compute the value of [tex]\( e^{0.25} \)[/tex]:
The mathematical constant [tex]\( e \)[/tex] raised to the power of 0.25 is:
[tex]\[
e^{0.25} \approx 1.284
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
Substituting the value of [tex]\( e^{0.25} \)[/tex], you get:
[tex]\[
P = \frac{288.9}{1.284} \approx 225
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 225. The correct answer is:
C. 225
