College

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. 1220
C. 50
D. 210

Answer :

To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex], we use the information that [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex].

1. Start with the given function:
[tex]\[ f(t) = P e^{rt} \][/tex]

2. Plug in the given values:
We have [tex]\( f(4) = 246.4 \)[/tex], [tex]\( r = 0.04 \)[/tex], and [tex]\( t = 4 \)[/tex]. Substitute these into the function:
[tex]\[ 246.4 = P e^{0.04 \times 4} \][/tex]

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

4. Solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{246.4}{e^{0.16}} \][/tex]

5. Calculate the approximate value:
After evaluating the expression [tex]\( \frac{246.4}{e^{0.16}} \)[/tex], we find that [tex]\( P \approx 210 \)[/tex].

Thus, the approximate value of [tex]\( P \)[/tex] is closest to option D: 210.