College

If [tex]$f(3) = 191.5$[/tex] when [tex]$r = 0.03$[/tex] for the function [tex]$f(t) = P e^{rt}$[/tex], then what is the approximate value of [tex][tex]$P$[/tex][/tex]?

A. 210
B. 78
C. 175
D. 471

Answer :

To find the value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^t \)[/tex] with the information given, we'll follow these steps:

1. We know that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].
2. The function is given as [tex]\( f(t) = P e^{rt} \)[/tex].

Let's plug in the known values:

- Since [tex]\( f(3) = 191.5 \)[/tex], we substitute 3 for [tex]\( t \)[/tex]:
[tex]\[
191.5 = P \cdot e^{0.03 \times 3}
\][/tex]

3. Next, simplify the exponent:
[tex]\[
e^{0.03 \times 3} = e^{0.09}
\][/tex]

4. To solve for [tex]\( P \)[/tex], rearrange the equation:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]

5. Now, calculate the approximate value of [tex]\( e^{0.09} \)[/tex] and divide 191.5 by this result to find [tex]\( P \)[/tex].

6. After calculating, the approximate value of [tex]\( P \)[/tex] is found to be 175.

Therefore, the correct option is:
C. 175