Answer :
To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex], when we know that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], we can follow these steps:
1. Substitute the known values into the function:
We have [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex]. Given [tex]\( f(5) = 288.9 \)[/tex], replace [tex]\( t \)[/tex] with 5 and [tex]\( r \)[/tex] with 0.05:
[tex]\[
288.9 = P \cdot e^{0.05 \times 5}
\][/tex]
2. Calculate the exponent:
Compute the exponent part, [tex]\( e^{0.05 \times 5} \)[/tex]:
[tex]\[
e^{0.25} \approx 1.284
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{1.284}
\][/tex]
4. Approximate the division:
Dividing [tex]\( 288.9 \)[/tex] by [tex]\( 1.284 \)[/tex] gives:
[tex]\[
P \approx 225
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 225, which corresponds to option D.
1. Substitute the known values into the function:
We have [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex]. Given [tex]\( f(5) = 288.9 \)[/tex], replace [tex]\( t \)[/tex] with 5 and [tex]\( r \)[/tex] with 0.05:
[tex]\[
288.9 = P \cdot e^{0.05 \times 5}
\][/tex]
2. Calculate the exponent:
Compute the exponent part, [tex]\( e^{0.05 \times 5} \)[/tex]:
[tex]\[
e^{0.25} \approx 1.284
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{1.284}
\][/tex]
4. Approximate the division:
Dividing [tex]\( 288.9 \)[/tex] by [tex]\( 1.284 \)[/tex] gives:
[tex]\[
P \approx 225
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 225, which corresponds to option D.
