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Answer :
To solve the problem, we need to find the value of [tex]\( P \)[/tex] when given the function [tex]\( f(t) = P \cdot e^{rt} \)[/tex]. We know that [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex].
Step-by-step solution:
1. Write down the given function and information:
- We're given the function: [tex]\( f(t) = P \cdot e^{rt} \)[/tex].
- Given data: [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex].
- We need to find [tex]\( P \)[/tex].
2. Substitute the known values into the function:
- We substitute [tex]\( t = 4 \)[/tex] and [tex]\( r = 0.04 \)[/tex] into the function:
[tex]\[
f(4) = P \cdot e^{0.04 \times 4}
\][/tex]
3. Use the given [tex]\( f(4) \)[/tex]:
- We know [tex]\( f(4) = 246.4 \)[/tex], so:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
- Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
5. Calculate [tex]\( e^{0.16} \)[/tex]:
- Calculate the value of [tex]\( e^{0.16} \)[/tex] approximately to find [tex]\( P \)[/tex].
6. Find [tex]\( P \)[/tex] and round to the closest answer choice:
- The calculation yields [tex]\( P \approx 209.968 \)[/tex].
- The closest whole number and given answer choice is 210.
Hence, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{210} \)[/tex].
Step-by-step solution:
1. Write down the given function and information:
- We're given the function: [tex]\( f(t) = P \cdot e^{rt} \)[/tex].
- Given data: [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex].
- We need to find [tex]\( P \)[/tex].
2. Substitute the known values into the function:
- We substitute [tex]\( t = 4 \)[/tex] and [tex]\( r = 0.04 \)[/tex] into the function:
[tex]\[
f(4) = P \cdot e^{0.04 \times 4}
\][/tex]
3. Use the given [tex]\( f(4) \)[/tex]:
- We know [tex]\( f(4) = 246.4 \)[/tex], so:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
- Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
5. Calculate [tex]\( e^{0.16} \)[/tex]:
- Calculate the value of [tex]\( e^{0.16} \)[/tex] approximately to find [tex]\( P \)[/tex].
6. Find [tex]\( P \)[/tex] and round to the closest answer choice:
- The calculation yields [tex]\( P \approx 209.968 \)[/tex].
- The closest whole number and given answer choice is 210.
Hence, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{210} \)[/tex].
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