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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], follow these steps:
1. Understand the function: We have the function [tex]\( f(t) = P e^t \)[/tex]. You need to determine the value of [tex]\( P \)[/tex].
2. Given values:
- [tex]\( f(5) = 288.9 \)[/tex].
- [tex]\( t = 5 \)[/tex].
3. Rearrange the equation: To find [tex]\( P \)[/tex], rearrange the function as:
[tex]\[
P = \frac{f(t)}{e^t}
\][/tex]
So, in our case:
[tex]\[
P = \frac{288.9}{e^5}
\][/tex]
4. Calculate [tex]\( e^5 \)[/tex]: Approximating [tex]\( e^5 \)[/tex] will give us a value about 148.41 (a common approximation).
5. Solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{148.41} \approx 1.95
\][/tex]
Based on this calculation, [tex]\( P \)[/tex] is approximately 1.95.
However, one of the answer choices provided must be selected. Given that 1.95 does not match any of the answer choices directly, the closest interpretation is that the problem or its solution might have inconsistencies or is meant to show a conceptual understanding rather than a matching of numbers to specific multiple choices. Nonetheless, the result derived is approximately 1.95 based on the given calculations.
1. Understand the function: We have the function [tex]\( f(t) = P e^t \)[/tex]. You need to determine the value of [tex]\( P \)[/tex].
2. Given values:
- [tex]\( f(5) = 288.9 \)[/tex].
- [tex]\( t = 5 \)[/tex].
3. Rearrange the equation: To find [tex]\( P \)[/tex], rearrange the function as:
[tex]\[
P = \frac{f(t)}{e^t}
\][/tex]
So, in our case:
[tex]\[
P = \frac{288.9}{e^5}
\][/tex]
4. Calculate [tex]\( e^5 \)[/tex]: Approximating [tex]\( e^5 \)[/tex] will give us a value about 148.41 (a common approximation).
5. Solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{148.41} \approx 1.95
\][/tex]
Based on this calculation, [tex]\( P \)[/tex] is approximately 1.95.
However, one of the answer choices provided must be selected. Given that 1.95 does not match any of the answer choices directly, the closest interpretation is that the problem or its solution might have inconsistencies or is meant to show a conceptual understanding rather than a matching of numbers to specific multiple choices. Nonetheless, the result derived is approximately 1.95 based on the given calculations.
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