Answer :
To solve the problem, we need to find the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{r t} \)[/tex], given the specific values: [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex].
Let's break this down step-by-step:
1. Understand the equation: The function is given as [tex]\( f(t) = P e^{r t} \)[/tex]. You need to rearrange this equation to solve for [tex]\( P \)[/tex].
2. Plug in the given values:
- We know that [tex]\( f(3) = 191.5 \)[/tex].
- Substitute [tex]\( t = 3 \)[/tex] and [tex]\( r = 0.03 \)[/tex] into the equation:
[tex]\[
191.5 = P e^{0.03 \times 3}
\][/tex]
3. Solve for [tex]\( e^{0.03 \times 3} \)[/tex]:
- Calculate the exponent: [tex]\( 0.03 \times 3 = 0.09 \)[/tex].
- Find [tex]\( e^{0.09} \)[/tex] using a calculator, which is approximately [tex]\( 1.094174 \)[/tex].
4. Solve for [tex]\( P \)[/tex]:
- Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
- Substitute the approximate value of [tex]\( e^{0.09} \)[/tex]:
[tex]\[
P = \frac{191.5}{1.094174} \approx 175.01782197944019
\][/tex]
5. Choose the closest answer:
- The value of [tex]\( P \)[/tex] is approximately 175.02.
- Compare this value to the options provided.
- The closest choice is A. 175.
Thus, the approximate value of [tex]\( P \)[/tex] is 175.
Let's break this down step-by-step:
1. Understand the equation: The function is given as [tex]\( f(t) = P e^{r t} \)[/tex]. You need to rearrange this equation to solve for [tex]\( P \)[/tex].
2. Plug in the given values:
- We know that [tex]\( f(3) = 191.5 \)[/tex].
- Substitute [tex]\( t = 3 \)[/tex] and [tex]\( r = 0.03 \)[/tex] into the equation:
[tex]\[
191.5 = P e^{0.03 \times 3}
\][/tex]
3. Solve for [tex]\( e^{0.03 \times 3} \)[/tex]:
- Calculate the exponent: [tex]\( 0.03 \times 3 = 0.09 \)[/tex].
- Find [tex]\( e^{0.09} \)[/tex] using a calculator, which is approximately [tex]\( 1.094174 \)[/tex].
4. Solve for [tex]\( P \)[/tex]:
- Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
- Substitute the approximate value of [tex]\( e^{0.09} \)[/tex]:
[tex]\[
P = \frac{191.5}{1.094174} \approx 175.01782197944019
\][/tex]
5. Choose the closest answer:
- The value of [tex]\( P \)[/tex] is approximately 175.02.
- Compare this value to the options provided.
- The closest choice is A. 175.
Thus, the approximate value of [tex]\( P \)[/tex] is 175.
