High School

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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. 210
C. 471
D. 175

Answer :

To solve this problem, we need to find the value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex] when given specific values. Here's how we can do that step-by-step:

1. Understand the Given Information:
We have [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].

2. Use the Function Formula:
The given function is [tex]\( f(t) = P e^{rt} \)[/tex].

3. Substitute the Known Values:
We know that [tex]\( t = 3 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( f(3) = 191.5 \)[/tex]. Substitute these into the function:
[tex]\[
191.5 = P \cdot e^{0.03 \times 3}
\][/tex]

4. Calculate the Exponential Part:
Calculate [tex]\( e^{0.03 \times 3} = e^{0.09} \)[/tex].

5. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]

6. Calculate the Value of [tex]\( P \)[/tex]:
Evaluate the right side of the equation to find the approximate value of [tex]\( P \)[/tex].

Using these steps, the approximate value of [tex]\( P \)[/tex] is very close to 175. Therefore, among the given options, the correct answer is:

D. 175

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