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Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex], we can use the given information [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex]. Let's go through the steps to solve for [tex]\( P \)[/tex].
1. Write down the function with the given values:
[tex]\[
f(4) = P \times e^{r \cdot t}
\][/tex]
Plugging in the values we have:
[tex]\[
246.4 = P \times e^{0.04 \times 4}
\][/tex]
2. Simplify the exponent:
[tex]\[
0.04 \times 4 = 0.16
\][/tex]
So the equation becomes:
[tex]\[
246.4 = P \times e^{0.16}
\][/tex]
3. Calculate [tex]\( e^{0.16} \)[/tex]. This is approximately equal to 1.17351.
4. Solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( e^{0.16} \)[/tex]:
[tex]\[
P = \frac{246.4}{1.17351}
\][/tex]
5. When you do the division:
[tex]\[
P \approx 209.97
\][/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is closest to 210. Therefore, the correct answer is:
D. 210
1. Write down the function with the given values:
[tex]\[
f(4) = P \times e^{r \cdot t}
\][/tex]
Plugging in the values we have:
[tex]\[
246.4 = P \times e^{0.04 \times 4}
\][/tex]
2. Simplify the exponent:
[tex]\[
0.04 \times 4 = 0.16
\][/tex]
So the equation becomes:
[tex]\[
246.4 = P \times e^{0.16}
\][/tex]
3. Calculate [tex]\( e^{0.16} \)[/tex]. This is approximately equal to 1.17351.
4. Solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( e^{0.16} \)[/tex]:
[tex]\[
P = \frac{246.4}{1.17351}
\][/tex]
5. When you do the division:
[tex]\[
P \approx 209.97
\][/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is closest to 210. Therefore, the correct answer is:
D. 210
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