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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. 175
B. 210
C. 471
D. 78

Answer :

To find the approximate value of [tex]\( P \)[/tex], we need to use the information given in the problem. We know that the function is [tex]\( f(t) = P e^{rt} \)[/tex].

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

Using the function format:

[tex]\[
f(3) = P e^{3r}
\][/tex]

Substituting the given values, we have:

[tex]\[
191.5 = P \times e^{3 \times 0.03}
\][/tex]

First, calculate the exponent:

[tex]\[
e^{3 \times 0.03} = e^{0.09} \approx 1.0941742837052104
\][/tex]

Now substitute this back into the equation:

[tex]\[
191.5 = P \times 1.0941742837052104
\][/tex]

To find [tex]\( P \)[/tex], solve for [tex]\( P \)[/tex]:

[tex]\[
P = \frac{191.5}{1.0941742837052104} \approx 175.01782197944019
\][/tex]

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

A. 175

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