Answer :
To find the value of [tex]\( P \)[/tex], we need to use the given function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex].
Here's the step-by-step solution:
1. Identify the Given Values:
- We know that [tex]\( f(3) = 191.5 \)[/tex].
- We are given [tex]\( r = 0.03 \)[/tex] and [tex]\( t = 3 \)[/tex].
2. Set Up the Equation:
[tex]\[
f(3) = P \cdot e^{0.03 \cdot 3}
\][/tex]
Substituting the given values:
[tex]\[
191.5 = P \cdot e^{0.09}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
- First, calculate [tex]\( e^{0.09} \)[/tex].
- This is approximately 1.0942.
Now substitute and solve for [tex]\( P \)[/tex]:
[tex]\[
191.5 = P \cdot 1.0942
\][/tex]
[tex]\[
P = \frac{191.5}{1.0942}
\][/tex]
4. Calculate the Value of [tex]\( P \)[/tex]:
[tex]\[
P \approx 175.02
\][/tex]
5. Choose the Closest Answer:
- The approximate value of [tex]\( P \)[/tex] is [tex]\( 175 \)[/tex].
Therefore, the correct answer is C. 175.
Here's the step-by-step solution:
1. Identify the Given Values:
- We know that [tex]\( f(3) = 191.5 \)[/tex].
- We are given [tex]\( r = 0.03 \)[/tex] and [tex]\( t = 3 \)[/tex].
2. Set Up the Equation:
[tex]\[
f(3) = P \cdot e^{0.03 \cdot 3}
\][/tex]
Substituting the given values:
[tex]\[
191.5 = P \cdot e^{0.09}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
- First, calculate [tex]\( e^{0.09} \)[/tex].
- This is approximately 1.0942.
Now substitute and solve for [tex]\( P \)[/tex]:
[tex]\[
191.5 = P \cdot 1.0942
\][/tex]
[tex]\[
P = \frac{191.5}{1.0942}
\][/tex]
4. Calculate the Value of [tex]\( P \)[/tex]:
[tex]\[
P \approx 175.02
\][/tex]
5. Choose the Closest Answer:
- The approximate value of [tex]\( P \)[/tex] is [tex]\( 175 \)[/tex].
Therefore, the correct answer is C. 175.
