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. 289
B. 1220
C. 50
D. 210

Answer :

To find the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{r t} \)[/tex], we use the information provided: [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex].

Here’s how you can find [tex]\( P \)[/tex] step by step:

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

2. Plug in the known values:
Since we know [tex]\( f(4) = 246.4 \)[/tex], [tex]\( r = 0.04 \)[/tex], and [tex]\( t = 4 \)[/tex], we substitute these values into the equation:
[tex]\[
246.4 = P e^{0.04 \times 4}
\][/tex]

3. Simplify the exponent:
Calculate the value of [tex]\( e^{0.04 \times 4} \)[/tex].
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]

4. Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735}
\][/tex]

5. Calculate [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]

6. Choose the closest approximate answer from the options:
The options are:
- A. 289
- B. 1220
- C. 50
- D. 210

The calculated value of [tex]\( P \approx 209.97 \)[/tex] is closest to option D: 210.

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

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