Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^t \)[/tex] where [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex], follow these steps:
1. Understand the Function and Given Values:
- The function is given as [tex]\( f(t) = P e^t \)[/tex].
- We know that when [tex]\( t = 4 \)[/tex], [tex]\( f(4) = 246.4 \)[/tex].
2. Identify the Required Calculation:
- We need to solve for [tex]\( P \)[/tex] using the formula [tex]\( f(t) = P e^t \)[/tex].
- Rearrange the formula to find [tex]\( P \)[/tex]:
[tex]\[
P = \frac{f(t)}{e^t}
\][/tex]
- Here, [tex]\( t = 4 \)[/tex] and [tex]\( f(4) = 246.4 \)[/tex].
3. Use the Base of Natural Logarithms [tex]\( e \)[/tex]:
- [tex]\( e \)[/tex] is a mathematical constant approximately equal to 2.71828.
4. Compute [tex]\( P \)[/tex]:
- Calculate the expression:
[tex]\[
P = \frac{246.4}{e^4}
\][/tex]
5. Result:
- The calculated value of [tex]\( P \)[/tex] is approximately [tex]\( 4.51 \)[/tex].
Since we are given specific answer choices, we should compare our result with the options:
- A. 210
- B. 289
- C. 1220
- D. 50
None of these options closely matches 4.51. Thus, it's likely there was a misunderstanding or mistake in the options. However, based on the calculations, the approximate value of [tex]\( P \)[/tex] is [tex]\( 4.51 \)[/tex], which doesn't fit the provided multiple-choice options.
1. Understand the Function and Given Values:
- The function is given as [tex]\( f(t) = P e^t \)[/tex].
- We know that when [tex]\( t = 4 \)[/tex], [tex]\( f(4) = 246.4 \)[/tex].
2. Identify the Required Calculation:
- We need to solve for [tex]\( P \)[/tex] using the formula [tex]\( f(t) = P e^t \)[/tex].
- Rearrange the formula to find [tex]\( P \)[/tex]:
[tex]\[
P = \frac{f(t)}{e^t}
\][/tex]
- Here, [tex]\( t = 4 \)[/tex] and [tex]\( f(4) = 246.4 \)[/tex].
3. Use the Base of Natural Logarithms [tex]\( e \)[/tex]:
- [tex]\( e \)[/tex] is a mathematical constant approximately equal to 2.71828.
4. Compute [tex]\( P \)[/tex]:
- Calculate the expression:
[tex]\[
P = \frac{246.4}{e^4}
\][/tex]
5. Result:
- The calculated value of [tex]\( P \)[/tex] is approximately [tex]\( 4.51 \)[/tex].
Since we are given specific answer choices, we should compare our result with the options:
- A. 210
- B. 289
- C. 1220
- D. 50
None of these options closely matches 4.51. Thus, it's likely there was a misunderstanding or mistake in the options. However, based on the calculations, the approximate value of [tex]\( P \)[/tex] is [tex]\( 4.51 \)[/tex], which doesn't fit the provided multiple-choice options.
