Answer :
To solve the problem, we need to find the value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex] given that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].
Let's break it down step-by-step:
1. Understand the Function:
The function is given as [tex]\( f(t) = P e^{rt} \)[/tex]. We know that when [tex]\( t = 3 \)[/tex], the function value is [tex]\( f(3) = 191.5 \)[/tex].
2. Substitute and Set Up the Equation:
Substitute the known values into the equation:
[tex]\[
191.5 = P \cdot e^{0.03 \times 3}
\][/tex]
This means:
[tex]\[
191.5 = P \cdot e^{0.09}
\][/tex]
3. Calculate [tex]\( e^{0.09} \)[/tex]:
Approximate the value of the exponential function:
[tex]\[
e^{0.09} \approx 1.0941742837052104
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{1.0941742837052104}
\][/tex]
[tex]\[
P \approx 175.01782197944019
\][/tex]
5. Choose the Closest Approximate Value:
The approximate value of [tex]\( P \)[/tex] is about 175.02, which is closest to option B: 175.
Therefore, the correct answer is B. 175.
Let's break it down step-by-step:
1. Understand the Function:
The function is given as [tex]\( f(t) = P e^{rt} \)[/tex]. We know that when [tex]\( t = 3 \)[/tex], the function value is [tex]\( f(3) = 191.5 \)[/tex].
2. Substitute and Set Up the Equation:
Substitute the known values into the equation:
[tex]\[
191.5 = P \cdot e^{0.03 \times 3}
\][/tex]
This means:
[tex]\[
191.5 = P \cdot e^{0.09}
\][/tex]
3. Calculate [tex]\( e^{0.09} \)[/tex]:
Approximate the value of the exponential function:
[tex]\[
e^{0.09} \approx 1.0941742837052104
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{1.0941742837052104}
\][/tex]
[tex]\[
P \approx 175.01782197944019
\][/tex]
5. Choose the Closest Approximate Value:
The approximate value of [tex]\( P \)[/tex] is about 175.02, which is closest to option B: 175.
Therefore, the correct answer is B. 175.
