College

If [tex]f(4) = 246.4[/tex] when [tex]r = 0.04[/tex] for the function [tex]f(t) = P e^t[/tex], then what is the approximate value of [tex]P[/tex]?

A. 1220
B. 50
C. 289
D. 210

Answer :

To solve for [tex]\( P \)[/tex] in the function [tex]\( f(t) = P \cdot e^{0.04t} \)[/tex] where [tex]\( f(4) = 246.4 \)[/tex], follow these steps:

1. We know the equation given is [tex]\( f(t) = P \cdot e^{0.04t} \)[/tex].

2. You're given that [tex]\( f(4) = 246.4 \)[/tex]. So, plug in [tex]\( t = 4 \)[/tex] into the function:
[tex]\[
f(4) = P \cdot e^{0.04 \times 4}
\][/tex]

3. Simplify the exponent:
[tex]\[
0.04 \times 4 = 0.16
\][/tex]

4. Now the equation becomes:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]

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

6. Calculate [tex]\( e^{0.16} \)[/tex] which is approximately equal to 1.1735.

7. Finally, solve for [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]

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