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Answer :
To solve for the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex], use the given information [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex].
1. First, write down the equation with the known values:
[tex]\[
f(4) = P \cdot e^{0.04 \times 4}
\][/tex]
2. Substitute the given value for [tex]\( f(4) \)[/tex]:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]
3. Calculate [tex]\( e^{0.16} \)[/tex]. From a calculation, we know:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
4. Substitute this value back into the equation:
[tex]\[
246.4 = P \cdot 1.1735
\][/tex]
5. Solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( 1.1735 \)[/tex]:
[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]
6. Round [tex]\( P \)[/tex] to the nearest whole number for the multiple-choice answer:
[tex]\[
P \approx 210
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 210. The correct answer is B. 210.
1. First, write down the equation with the known values:
[tex]\[
f(4) = P \cdot e^{0.04 \times 4}
\][/tex]
2. Substitute the given value for [tex]\( f(4) \)[/tex]:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]
3. Calculate [tex]\( e^{0.16} \)[/tex]. From a calculation, we know:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
4. Substitute this value back into the equation:
[tex]\[
246.4 = P \cdot 1.1735
\][/tex]
5. Solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( 1.1735 \)[/tex]:
[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]
6. Round [tex]\( P \)[/tex] to the nearest whole number for the multiple-choice answer:
[tex]\[
P \approx 210
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 210. The correct answer is B. 210.
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