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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. 210
D. 50

Answer :

Sure! To find the approximate value of [tex]\( P \)[/tex], we are given a function [tex]\( f(t) = P e^{rt} \)[/tex] where [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex]. Let's break down the steps to solve for [tex]\( P \)[/tex]:

1. Set up the equation with the given information:

We know:
[tex]\[
f(4) = 246.4
\][/tex]
So,
[tex]\[
246.4 = P \cdot e^{0.04 \times 4}
\][/tex]

2. Calculate the value of the exponent:

First, find the exponent:
[tex]\[
0.04 \times 4 = 0.16
\][/tex]

3. Calculate [tex]\( e^{0.16} \)[/tex]:

Using a calculator, compute:
[tex]\[
e^{0.16} \approx 1.17351
\][/tex]

4. Solve for [tex]\( P \)[/tex]:

Substitute the value of [tex]\( e^{0.16} \)[/tex] into the equation:
[tex]\[
246.4 = P \cdot 1.17351
\][/tex]
To isolate [tex]\( P \)[/tex], divide both sides by [tex]\( 1.17351 \)[/tex]:
[tex]\[
P = \frac{246.4}{1.17351}
\][/tex]

5. Perform the division:

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

The approximate value of [tex]\( P \)[/tex] is closest to 210. Therefore, the correct answer is:

C. 210

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