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Answer :
To solve for the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{r \cdot t} \)[/tex], where [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], follow these steps:
1. Identify the Known Values: We know that:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]
2. Write Down the Function: The function provided is [tex]\( f(t) = P e^{r \cdot t} \)[/tex]. Substitute the given values into this function:
[tex]\[
f(5) = P e^{0.05 \cdot 5}
\][/tex]
3. Substitute the Known Value of [tex]\( f(5) \)[/tex]: Replace [tex]\( f(5) \)[/tex] with 288.9:
[tex]\[
288.9 = P e^{0.25}
\][/tex]
4. Solve for [tex]\( P \)[/tex]: To find [tex]\( P \)[/tex], divide both sides of the equation by [tex]\( e^{0.25} \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
5. Approximate the Value of [tex]\( e^{0.25} \)[/tex]: Calculate [tex]\( e^{0.25} \)[/tex] using the exponential function, which gives approximately 1.284.
6. Calculate [tex]\( P \)[/tex]: Now substitute this value back into the equation:
[tex]\[
P \approx \frac{288.9}{1.284} \approx 225
\][/tex]
Based on the calculations, the approximate value of [tex]\( P \)[/tex] is closest to 225. So, the correct choice is:
A. 225
1. Identify the Known Values: We know that:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]
2. Write Down the Function: The function provided is [tex]\( f(t) = P e^{r \cdot t} \)[/tex]. Substitute the given values into this function:
[tex]\[
f(5) = P e^{0.05 \cdot 5}
\][/tex]
3. Substitute the Known Value of [tex]\( f(5) \)[/tex]: Replace [tex]\( f(5) \)[/tex] with 288.9:
[tex]\[
288.9 = P e^{0.25}
\][/tex]
4. Solve for [tex]\( P \)[/tex]: To find [tex]\( P \)[/tex], divide both sides of the equation by [tex]\( e^{0.25} \)[/tex]:
[tex]\[
P = \frac{288.9}{e^{0.25}}
\][/tex]
5. Approximate the Value of [tex]\( e^{0.25} \)[/tex]: Calculate [tex]\( e^{0.25} \)[/tex] using the exponential function, which gives approximately 1.284.
6. Calculate [tex]\( P \)[/tex]: Now substitute this value back into the equation:
[tex]\[
P \approx \frac{288.9}{1.284} \approx 225
\][/tex]
Based on the calculations, the approximate value of [tex]\( P \)[/tex] is closest to 225. So, the correct choice is:
A. 225
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