High School

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. 289
D. 1220

Answer :

To solve this problem, we need to find the value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex]. It's given that [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex].

Let's break down the steps:

1. Identify the given values:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]

2. Rewrite the function:
The function is rewritten using the values [tex]\( r \)[/tex] and [tex]\( t \)[/tex]:
[tex]\[
f(t) = P \cdot e^{r \cdot t}
\][/tex]
Substituting the given values:
[tex]\[
f(4) = P \cdot e^{0.04 \cdot 4}
\][/tex]

3. Set up the equation:
Substitute [tex]\( f(4) = 246.4 \)[/tex] into the equation:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]

4. Solve for [tex]\( P \)[/tex]:
To solve for [tex]\( P \)[/tex], we need to divide both sides by [tex]\( e^{0.16} \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]

5. Calculate the approximate value of [tex]\( P \)[/tex]:
By evaluating [tex]\( e^{0.16} \)[/tex] and dividing, we find that:
[tex]\[
P \approx 209.97
\][/tex]

Since the value of [tex]\( P \approx 209.97 \)[/tex] closely matches option A, which is 210, the approximate value of [tex]\( P \)[/tex] is:

A. 210

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