Answer :
To solve the problem, we need to find the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex] where [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex].
Here's a step-by-step approach:
1. Understand the given equation:
The function is given as [tex]\( f(t) = P e^{rt} \)[/tex]. We know that [tex]\( f(4) = 246.4 \)[/tex].
2. Substitute the known values:
When [tex]\( t = 4 \)[/tex], the equation becomes:
[tex]\[
246.4 = P \times e^{(4 \times 0.04)}
\][/tex]
3. Simplify the exponent:
Calculate the exponent in [tex]\( e^{(4 \times 0.04)}\)[/tex]:
[tex]\[
4 \times 0.04 = 0.16
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
5. Calculate [tex]\( e^{0.16} \)[/tex]:
Using the given result, we calculate the value:
[tex]\[
e^{0.16} \approx 1.17351
\][/tex]
6. Find the approximate value of [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.17351} \approx 209.97
\][/tex]
7. Choose the closest option:
The approximate value of [tex]\( P \)[/tex] is closest to 210.
Therefore, the answer is [tex]\( \boxed{210} \)[/tex].
Here's a step-by-step approach:
1. Understand the given equation:
The function is given as [tex]\( f(t) = P e^{rt} \)[/tex]. We know that [tex]\( f(4) = 246.4 \)[/tex].
2. Substitute the known values:
When [tex]\( t = 4 \)[/tex], the equation becomes:
[tex]\[
246.4 = P \times e^{(4 \times 0.04)}
\][/tex]
3. Simplify the exponent:
Calculate the exponent in [tex]\( e^{(4 \times 0.04)}\)[/tex]:
[tex]\[
4 \times 0.04 = 0.16
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
5. Calculate [tex]\( e^{0.16} \)[/tex]:
Using the given result, we calculate the value:
[tex]\[
e^{0.16} \approx 1.17351
\][/tex]
6. Find the approximate value of [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.17351} \approx 209.97
\][/tex]
7. Choose the closest option:
The approximate value of [tex]\( P \)[/tex] is closest to 210.
Therefore, the answer is [tex]\( \boxed{210} \)[/tex].
