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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. 1220
B. 210
C. 50
D. 289

Answer :

Sure! Let's solve the problem step-by-step to find the approximate value of [tex]\( P \)[/tex].

We are given a function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex], where:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]

We need to find the initial value [tex]\( P \)[/tex].

1. Substitute the known values into the equation:

[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]

2. Calculate the exponent:

The exponent term [tex]\( 0.04 \cdot 4 = 0.16 \)[/tex], so we need to determine [tex]\( e^{0.16} \)[/tex].

3. Rearrange the equation to solve for [tex]\( P \)[/tex]:

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

4. Calculate the approximate value of [tex]\( e^{0.16} \)[/tex]:

Using a calculator, find [tex]\( e^{0.16} \approx 1.17351 \)[/tex].

5. Divide to find [tex]\( P \)[/tex]:

[tex]\[
P \approx \frac{246.4}{1.17351} \approx 210
\][/tex]

6. Select the closest multiple-choice answer:

The approximate value of [tex]\( P \)[/tex] is around 210.

Therefore, the correct choice is:
- B. 210

That's the step-by-step explanation for finding the approximate value of [tex]\( P \)[/tex].

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