If [tex]f(4) = 246.4[/tex] when [tex]r = 0.04[/tex] for the function [tex]f(t) = P e^{rt}[/tex], then what is the approximate value of [tex]P[/tex]?

A. 210
B. 50
C. 1220
D. 289



Answer :

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].