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

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

Answer :

Let's solve the problem step-by-step to find the value of [tex]\( P \)[/tex] in the equation [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex].

We are given:
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]

The function is:
[tex]\[ f(t) = P \cdot e^{r \cdot t} \][/tex]

In this case:
[tex]\[ f(3) = P \cdot e^{0.03 \times 3} = 191.5 \][/tex]

Now, let's calculate [tex]\( e^{0.03 \times 3} \)[/tex]:
[tex]\[ e^{0.09} \approx 1.0942 \][/tex] (This is a standard exponentiation result; you can use a calculator to verify.)

Now substitute back to solve for [tex]\( P \)[/tex]:
[tex]\[ 191.5 = P \cdot 1.0942 \][/tex]

To find [tex]\( P \)[/tex], divide both sides by 1.0942:
[tex]\[ P = \frac{191.5}{1.0942} \][/tex]

Calculating this gives:
[tex]\[ P \approx 175 \][/tex]

Therefore, the approximate value of [tex]\( P \)[/tex] is 175.

The correct answer is D. 175.