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Answer :
To solve for the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P \cdot e^t \)[/tex], follow these steps:
1. Identify the Given Values:
- We're told that [tex]\( f(4) = 246.4 \)[/tex].
- The interest rate, [tex]\( r \)[/tex], is 0.04. However, note that this value of [tex]\( r \)[/tex] doesn't directly impact our calculation since [tex]\( t = 4 \)[/tex] is already given separately.
2. Understand the Function:
- The function is [tex]\( f(t) = P \cdot e^t \)[/tex]. We have [tex]\( f(4) = 246.4 \)[/tex], so we can express it as:
[tex]\[
246.4 = P \cdot e^4
\][/tex]
3. Rearrange the Equation:
- To solve for [tex]\( P \)[/tex], rearrange the equation to isolate [tex]\( P \)[/tex]:
[tex]\[
P = \frac{f(4)}{e^4}
\][/tex]
4. Calculate the Required Expressions:
- Calculate [tex]\( e^4 \)[/tex]. Through accurate calculation, we find that [tex]\( e^4 \approx 54.60 \)[/tex].
5. Compute [tex]\( P \)[/tex]:
- Substitute the values into the equation for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{54.60} \approx 4.51
\][/tex]
Since none of the given options (A. 50, B. 289, C. 1220, D. 210) are close to 4.51, it seems there might be a mistake in the context or choices provided. But according to our calculation, the approximate value of [tex]\( P \)[/tex] when rounded is about 4.5.
1. Identify the Given Values:
- We're told that [tex]\( f(4) = 246.4 \)[/tex].
- The interest rate, [tex]\( r \)[/tex], is 0.04. However, note that this value of [tex]\( r \)[/tex] doesn't directly impact our calculation since [tex]\( t = 4 \)[/tex] is already given separately.
2. Understand the Function:
- The function is [tex]\( f(t) = P \cdot e^t \)[/tex]. We have [tex]\( f(4) = 246.4 \)[/tex], so we can express it as:
[tex]\[
246.4 = P \cdot e^4
\][/tex]
3. Rearrange the Equation:
- To solve for [tex]\( P \)[/tex], rearrange the equation to isolate [tex]\( P \)[/tex]:
[tex]\[
P = \frac{f(4)}{e^4}
\][/tex]
4. Calculate the Required Expressions:
- Calculate [tex]\( e^4 \)[/tex]. Through accurate calculation, we find that [tex]\( e^4 \approx 54.60 \)[/tex].
5. Compute [tex]\( P \)[/tex]:
- Substitute the values into the equation for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{54.60} \approx 4.51
\][/tex]
Since none of the given options (A. 50, B. 289, C. 1220, D. 210) are close to 4.51, it seems there might be a mistake in the context or choices provided. But according to our calculation, the approximate value of [tex]\( P \)[/tex] when rounded is about 4.5.
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