To determine the initial amount [tex]\( P \)[/tex] from the given function [tex]\( f(t) = P e^{rt} \)[/tex] when certain parameters are provided, we need to follow a few steps.
The function in this context is:
[tex]\[ f(t) = P e^{rt} \][/tex]
Given:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
We are required to approximate the value of [tex]\( P \)[/tex].
Step-by-Step Solution:
1. Identify the given values:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
2. Write the equation with the known values:
[tex]\[ 246.4 = P \cdot e^{0.04 \cdot 4} \][/tex]
3. Simplify the exponent:
- Calculate the exponent: [tex]\( 0.04 \times 4 = 0.16 \)[/tex]
4. Express the function in terms of the exponent:
[tex]\[ 246.4 = P \cdot e^{0.16} \][/tex]
5. Calculate [tex]\( e^{0.16} \)[/tex]:
- Approximately: [tex]\( e^{0.16} \approx 1.1735108709918103 \)[/tex]
6. Rearrange to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{246.4}{1.1735108709918103} \][/tex]
7. Calculate [tex]\( P \)[/tex]:
[tex]\[ P \approx 209.96822960127446 \][/tex]
Given the multiple-choice options:
A. 50
B. 210
C. 1220
D. 289
The calculated value of [tex]\( P \approx 209.97 \)[/tex] is closest to option B.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( 210 \)[/tex].
