Answer :
To find the approximate value of [tex]\( P \)[/tex], given the equation [tex]\( f(t) = P \cdot e^{rt} \)[/tex] where [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], we can follow these steps:
1. Identify the components: We know [tex]\( f(5) = 288.9 \)[/tex], [tex]\( r = 0.05 \)[/tex], and we assume [tex]\( t = 5 \)[/tex].
2. Substitute known values into the function: Substitute the known values into the equation [tex]\( f(t) = P \cdot e^{rt} \)[/tex]. So, it becomes:
[tex]\[
288.9 = P \cdot e^{0.05 \times 5}
\][/tex]
3. Calculate [tex]\( e^{0.05 \times 5} \)[/tex]: First, compute the exponent:
[tex]\[
e^{0.25} \approx 1.284025
\][/tex]
4. Solve for [tex]\( P \)[/tex]: Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{1.284025}
\][/tex]
5. Perform the division:
[tex]\[
P \approx 224.99554622932885
\][/tex]
6. Approximate [tex]\( P \)[/tex] to the nearest answer choice: Comparing the computed value of [tex]\( P \)[/tex] to the given options, it is closest to:
- [tex]\( \text{D. } 225 \)[/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is 225.
1. Identify the components: We know [tex]\( f(5) = 288.9 \)[/tex], [tex]\( r = 0.05 \)[/tex], and we assume [tex]\( t = 5 \)[/tex].
2. Substitute known values into the function: Substitute the known values into the equation [tex]\( f(t) = P \cdot e^{rt} \)[/tex]. So, it becomes:
[tex]\[
288.9 = P \cdot e^{0.05 \times 5}
\][/tex]
3. Calculate [tex]\( e^{0.05 \times 5} \)[/tex]: First, compute the exponent:
[tex]\[
e^{0.25} \approx 1.284025
\][/tex]
4. Solve for [tex]\( P \)[/tex]: Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{1.284025}
\][/tex]
5. Perform the division:
[tex]\[
P \approx 224.99554622932885
\][/tex]
6. Approximate [tex]\( P \)[/tex] to the nearest answer choice: Comparing the computed value of [tex]\( P \)[/tex] to the given options, it is closest to:
- [tex]\( \text{D. } 225 \)[/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is 225.
