To determine the approximate value of [tex]\( P \)[/tex], given the function [tex]\( f(t) = P e^{rt} \)[/tex] and the conditions [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex], we proceed as follows:
1. Identify the given values and function:
- Function: [tex]\( f(t) = P e^{rt} \)[/tex]
- Condition 1: [tex]\( f(4) = 246.4 \)[/tex]
- Condition 2: [tex]\( r = 0.04 \)[/tex]
2. Substitute the given values into the function:
[tex]\[
246.4 = P e^{0.04 \cdot 4}
\][/tex]
3. Simplify the exponent:
[tex]\[
246.4 = P e^{0.16}
\][/tex]
4. Calculate [tex]\( e^{0.16} \)[/tex]:
[tex]\[
e^{0.16} \approx 1.17351
\][/tex]
5. Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.17351}
\][/tex]
6. Perform the division:
[tex]\[
P \approx 209.96822960127446
\][/tex]
7. Match the approximate value of [tex]\( P \)[/tex] with the closest choice from the given options:
- Choices given: [tex]\( 210, 50, 1220, 289 \)[/tex]
- The closest value to [tex]\( 209.96822960127446 \)[/tex] is [tex]\( 210 \)[/tex].
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\(\boxed{210}\)[/tex].
