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Answer :
To solve the problem, we are given that for a function [tex]\( f(t) = P e^t \)[/tex], the value of [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex]. We need to find the approximate value of [tex]\( P \)[/tex].
Here's how we can approach this step-by-step:
1. Understand the given function:
The function is [tex]\( f(t) = P e^t \)[/tex]. Here, [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
2. Plug in the given values:
For [tex]\( t = 4 \)[/tex], we are given [tex]\( f(4) = 246.4 \)[/tex].
3. Set up the equation:
[tex]\[
f(4) = P e^4 = 246.4
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
To find [tex]\( P \)[/tex], we need to rearrange the equation:
[tex]\[
P = \frac{f(4)}{e^4}
\][/tex]
5. Calculate [tex]\( e^4 \)[/tex]:
The value of [tex]\( e^4 \)[/tex] is approximately 54.598.
6. Calculate [tex]\( P \)[/tex]:
Substitute [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( e^4 = 54.598 \)[/tex] into the equation for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{54.598} \approx 4.51
\][/tex]
The possible options are:
- A. 1220
- B. 50
- C. 210
- D. 289
Since our calculated [tex]\( P \)[/tex] value (approximately 4.51) doesn't match any of the options directly, it seems there might be some misunderstanding or a different interpretation of the options. However, based on our calculation, the approximate value of [tex]\( P \)[/tex] would be closest to 4.51.
Here's how we can approach this step-by-step:
1. Understand the given function:
The function is [tex]\( f(t) = P e^t \)[/tex]. Here, [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
2. Plug in the given values:
For [tex]\( t = 4 \)[/tex], we are given [tex]\( f(4) = 246.4 \)[/tex].
3. Set up the equation:
[tex]\[
f(4) = P e^4 = 246.4
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
To find [tex]\( P \)[/tex], we need to rearrange the equation:
[tex]\[
P = \frac{f(4)}{e^4}
\][/tex]
5. Calculate [tex]\( e^4 \)[/tex]:
The value of [tex]\( e^4 \)[/tex] is approximately 54.598.
6. Calculate [tex]\( P \)[/tex]:
Substitute [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( e^4 = 54.598 \)[/tex] into the equation for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{54.598} \approx 4.51
\][/tex]
The possible options are:
- A. 1220
- B. 50
- C. 210
- D. 289
Since our calculated [tex]\( P \)[/tex] value (approximately 4.51) doesn't match any of the options directly, it seems there might be some misunderstanding or a different interpretation of the options. However, based on our calculation, the approximate value of [tex]\( P \)[/tex] would be closest to 4.51.
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