College

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. 50
B. 210
C. 289
D. 1220

Answer :

Sure, let's solve this problem step by step.

We're given the function [tex]\( f(t) = P e^{rt} \)[/tex], and we know:

- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]

We're asked to find the approximate value of [tex]\( P \)[/tex].

1. Write down the equation: The function given is [tex]\( f(t) = P e^{rt} \)[/tex].

2. Substitute the known values: We need to find [tex]\( P \)[/tex] using the equation [tex]\( f(4) = P e^{0.04 \times 4} \)[/tex].

This becomes:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]

3. Solve for [tex]\( P \)[/tex]: To find [tex]\( P \)[/tex], rearrange the equation:

[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]

4. Calculate [tex]\( e^{0.16} \)[/tex]: The approximate value of [tex]\( e^{0.16} \)[/tex] is used in calculating [tex]\( P \)[/tex].

5. Divide to find [tex]\( P \)[/tex]: Substitute the value of [tex]\( e^{0.16} \)[/tex] to solve for [tex]\( P \)[/tex].

[tex]\[
P \approx 209.97
\][/tex]

Therefore, after solving, the approximate value of [tex]\( P \)[/tex] is closest to 210.

The correct answer is B. 210.

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