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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. 210
B. 78
C. 175
D. 471

Answer :

To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{r \cdot t} \)[/tex], we start with the information provided:

- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- [tex]\( t = 3 \)[/tex]

The function at [tex]\( t = 3 \)[/tex] is given by:
[tex]\[ f(3) = P \cdot e^{r \cdot 3} \][/tex]

So we have the equation:
[tex]\[ 191.5 = P \cdot e^{0.03 \cdot 3} \][/tex]

Step 1: Calculate the exponent in the expression [tex]\( e^{0.03 \cdot 3} \)[/tex]:

[tex]\[ r \cdot t = 0.03 \times 3 = 0.09 \][/tex]

Step 2: Calculate [tex]\( e^{0.09} \)[/tex]:

This evaluates to approximately 1.0942.

Step 3: Solve for [tex]\( P \)[/tex]:

Rearranging the equation to solve for [tex]\( P \)[/tex], we get:
[tex]\[ P = \frac{191.5}{e^{0.09}} \][/tex]

Plug in the value of [tex]\( e^{0.09} \)[/tex]:
[tex]\[ P \approx \frac{191.5}{1.0942} \][/tex]

[tex]\[ P \approx 175.02 \][/tex]

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

So, the correct answer is C. 175.

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