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Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P \times e^{r \times t} \)[/tex] where [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], follow these steps:
1. Identify the given information:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]
2. Substitute the given values into the function formula:
[tex]\[
288.9 = P \times e^{0.05 \times 5}
\][/tex]
3. Calculate the exponent:
[tex]\[
e^{0.05 \times 5} = e^{0.25}
\][/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]
5. Calculate [tex]\( e^{0.25} \)[/tex]:
- Using the given accurate calculations, [tex]\( e^{0.25} \approx 1.2840254166877414 \)[/tex].
6. Plug the value of [tex]\( e^{0.25} \)[/tex] back into the equation and divide:
[tex]\[
P = \frac{288.9}{1.2840254166877414} \approx 224.99554622932885
\][/tex]
7. Round the value to the closest option:
- The approximate value of [tex]\( P \)[/tex] is [tex]\( 225 \)[/tex].
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{225} \)[/tex], which corresponds to option C.
1. Identify the given information:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]
2. Substitute the given values into the function formula:
[tex]\[
288.9 = P \times e^{0.05 \times 5}
\][/tex]
3. Calculate the exponent:
[tex]\[
e^{0.05 \times 5} = e^{0.25}
\][/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]
5. Calculate [tex]\( e^{0.25} \)[/tex]:
- Using the given accurate calculations, [tex]\( e^{0.25} \approx 1.2840254166877414 \)[/tex].
6. Plug the value of [tex]\( e^{0.25} \)[/tex] back into the equation and divide:
[tex]\[
P = \frac{288.9}{1.2840254166877414} \approx 224.99554622932885
\][/tex]
7. Round the value to the closest option:
- The approximate value of [tex]\( P \)[/tex] is [tex]\( 225 \)[/tex].
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{225} \)[/tex], which corresponds to option C.
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