Answer :
To find the approximate value of [tex]\( P \)[/tex], we need to use the given function [tex]\( f(t) = P e^{rt} \)[/tex] and the provided information that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].
### Steps:
1. Understand the Problem:
- We have a function: [tex]\( f(t) = P e^{rt} \)[/tex].
- Given: [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex].
2. Set Up the Equation:
- Substitute the given values into the equation:
[tex]\[
f(3) = P e^{0.03 \times 3}
\][/tex]
- This simplifies to:
[tex]\[
191.5 = P e^{0.09}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
- To find [tex]\( P \)[/tex], rearrange the equation:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
- Calculate [tex]\( e^{0.09} \)[/tex] using a calculator. It’s approximately 1.09417.
4. Calculate [tex]\( P \)[/tex]:
- Plug this value back to find [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{191.5}{1.09417} \approx 175.02
\][/tex]
5. Round to a Provided Option:
- The closest value in the options provided is 175.
Thus, the approximate value of [tex]\( P \)[/tex] is 175. The correct answer is option C.
### Steps:
1. Understand the Problem:
- We have a function: [tex]\( f(t) = P e^{rt} \)[/tex].
- Given: [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex].
2. Set Up the Equation:
- Substitute the given values into the equation:
[tex]\[
f(3) = P e^{0.03 \times 3}
\][/tex]
- This simplifies to:
[tex]\[
191.5 = P e^{0.09}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
- To find [tex]\( P \)[/tex], rearrange the equation:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
- Calculate [tex]\( e^{0.09} \)[/tex] using a calculator. It’s approximately 1.09417.
4. Calculate [tex]\( P \)[/tex]:
- Plug this value back to find [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{191.5}{1.09417} \approx 175.02
\][/tex]
5. Round to a Provided Option:
- The closest value in the options provided is 175.
Thus, the approximate value of [tex]\( P \)[/tex] is 175. The correct answer is option C.
