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Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{r t} \)[/tex], where we know that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex], we can follow these steps:
1. Write down the expression for [tex]\( f(t) \)[/tex]:
[tex]\[
f(t) = P e^{r t}
\][/tex]
2. Substitute the given values into the expression:
[tex]\[
f(3) = P e^{0.03 \times 3}
\][/tex]
We know [tex]\( f(3) = 191.5 \)[/tex].
3. Set up the equation with known values:
[tex]\[
191.5 = P e^{0.09}
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
We need to isolate [tex]\( P \)[/tex], so divide both sides by [tex]\( e^{0.09} \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Calculate [tex]\( e^{0.09} \)[/tex]:
Without manually computing the exponential, we use the given result that [tex]\( \frac{191.5}{e^{0.09}} \approx 175.01782197944019 \)[/tex].
Using these steps, the approximate value of [tex]\( P \)[/tex] is:
C. 175
1. Write down the expression for [tex]\( f(t) \)[/tex]:
[tex]\[
f(t) = P e^{r t}
\][/tex]
2. Substitute the given values into the expression:
[tex]\[
f(3) = P e^{0.03 \times 3}
\][/tex]
We know [tex]\( f(3) = 191.5 \)[/tex].
3. Set up the equation with known values:
[tex]\[
191.5 = P e^{0.09}
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
We need to isolate [tex]\( P \)[/tex], so divide both sides by [tex]\( e^{0.09} \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Calculate [tex]\( e^{0.09} \)[/tex]:
Without manually computing the exponential, we use the given result that [tex]\( \frac{191.5}{e^{0.09}} \approx 175.01782197944019 \)[/tex].
Using these steps, the approximate value of [tex]\( P \)[/tex] is:
C. 175
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