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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. 50
B. 1220
C. 210
D. 289

Answer :

Let's work through the solution step-by-step to find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex].

1. Given Information:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]

2. Function Formula:
The function is given by:
[tex]\[
f(t) = P e^{rt}
\][/tex]

3. Set Up the Equation:
Substitute the known values into the function to find [tex]\( P \)[/tex]:
[tex]\[
f(4) = P e^{0.04 \times 4}
\][/tex]
[tex]\[
246.4 = P e^{0.16}
\][/tex]

4. Calculate [tex]\( e^{0.16} \)[/tex]:
Use the exponential function [tex]\( e^{0.16} \)[/tex] to find its value:
[tex]\[
e^{0.16} \approx 1.17351
\][/tex]

5. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.17351}
\][/tex]
[tex]\[
P \approx 209.97
\][/tex]

6. Approximate [tex]\( P \)[/tex]:
Round [tex]\( P \)[/tex] to the nearest whole number and see which option it closely matches:
[tex]\[
P \approx 210
\][/tex]

Therefore, the approximate value of [tex]\( P \)[/tex] is 210, which corresponds to option C.

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