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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. 50
C. 1220
D. 210

Answer :

To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex], we are given that [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex].

Step-by-step, we'll solve for [tex]\( P \)[/tex] as follows:

1. Identify the given data:
- For [tex]\( t = 4 \)[/tex], [tex]\( f(4) = 246.4 \)[/tex].
- The rate [tex]\( r = 0.04 \)[/tex].

2. Write the function equation:
[tex]\[
f(t) = P e^{rt}
\][/tex]

3. Substitute the known values into the equation:
[tex]\[
246.4 = P e^{(0.04 \times 4)}
\][/tex]

4. Calculate the exponent:
[tex]\[
e^{(0.04 \times 4)} = e^{0.16}
\][/tex]

5. Estimate [tex]\( e^{0.16} \)[/tex]:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]

6. Solve for [tex]\( P \)[/tex]:
[tex]\[
246.4 = P \times 1.1735
\][/tex]

7. Divide both sides by [tex]\( 1.1735 \)[/tex] to find [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]

Based on this calculation, the approximate value of [tex]\( P \)[/tex] is closest to option D, 210.

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