Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{r \cdot t} \)[/tex], we start with the information provided:
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- [tex]\( t = 3 \)[/tex]
The function at [tex]\( t = 3 \)[/tex] is given by:
[tex]\[ f(3) = P \cdot e^{r \cdot 3} \][/tex]
So we have the equation:
[tex]\[ 191.5 = P \cdot e^{0.03 \cdot 3} \][/tex]
Step 1: Calculate the exponent in the expression [tex]\( e^{0.03 \cdot 3} \)[/tex]:
[tex]\[ r \cdot t = 0.03 \times 3 = 0.09 \][/tex]
Step 2: Calculate [tex]\( e^{0.09} \)[/tex]:
This evaluates to approximately 1.0942.
Step 3: Solve for [tex]\( P \)[/tex]:
Rearranging the equation to solve for [tex]\( P \)[/tex], we get:
[tex]\[ P = \frac{191.5}{e^{0.09}} \][/tex]
Plug in the value of [tex]\( e^{0.09} \)[/tex]:
[tex]\[ P \approx \frac{191.5}{1.0942} \][/tex]
[tex]\[ P \approx 175.02 \][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 175.
So, the correct answer is C. 175.
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- [tex]\( t = 3 \)[/tex]
The function at [tex]\( t = 3 \)[/tex] is given by:
[tex]\[ f(3) = P \cdot e^{r \cdot 3} \][/tex]
So we have the equation:
[tex]\[ 191.5 = P \cdot e^{0.03 \cdot 3} \][/tex]
Step 1: Calculate the exponent in the expression [tex]\( e^{0.03 \cdot 3} \)[/tex]:
[tex]\[ r \cdot t = 0.03 \times 3 = 0.09 \][/tex]
Step 2: Calculate [tex]\( e^{0.09} \)[/tex]:
This evaluates to approximately 1.0942.
Step 3: Solve for [tex]\( P \)[/tex]:
Rearranging the equation to solve for [tex]\( P \)[/tex], we get:
[tex]\[ P = \frac{191.5}{e^{0.09}} \][/tex]
Plug in the value of [tex]\( e^{0.09} \)[/tex]:
[tex]\[ P \approx \frac{191.5}{1.0942} \][/tex]
[tex]\[ P \approx 175.02 \][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 175.
So, the correct answer is C. 175.
