Answer :
To find the approximate value of [tex]\( P \)[/tex], we need to solve the equation given by the function [tex]\( f(t) = P e^{r \cdot t} \)[/tex].
We're given:
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- [tex]\( t = 3 \)[/tex]
The function can be rewritten for these values as:
[tex]\[ 191.5 = P e^{0.03 \cdot 3} \][/tex]
We need to solve for [tex]\( P \)[/tex]. Follow these steps:
1. Calculate the Exponent:
First, calculate the exponent [tex]\( r \cdot t \)[/tex]:
[tex]\[ r \cdot t = 0.03 \times 3 = 0.09 \][/tex]
2. Find [tex]\( e^{0.09} \)[/tex]:
Calculate the value of [tex]\( e^{0.09} \)[/tex]. (Note: This value is calculated using a scientific calculator and typically, [tex]\( e^{0.09} \approx 1.09417 \)[/tex].)
3. Rearrange the Equation:
Substitute [tex]\( e^{0.09} \)[/tex] back into the equation:
[tex]\[ 191.5 = P \times 1.09417 \][/tex]
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{191.5}{1.09417} \][/tex]
5. Calculate [tex]\( P \)[/tex]:
The division results in:
[tex]\[ P \approx 175.0178 \][/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is 175. So, the correct answer is:
B. 175
We're given:
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- [tex]\( t = 3 \)[/tex]
The function can be rewritten for these values as:
[tex]\[ 191.5 = P e^{0.03 \cdot 3} \][/tex]
We need to solve for [tex]\( P \)[/tex]. Follow these steps:
1. Calculate the Exponent:
First, calculate the exponent [tex]\( r \cdot t \)[/tex]:
[tex]\[ r \cdot t = 0.03 \times 3 = 0.09 \][/tex]
2. Find [tex]\( e^{0.09} \)[/tex]:
Calculate the value of [tex]\( e^{0.09} \)[/tex]. (Note: This value is calculated using a scientific calculator and typically, [tex]\( e^{0.09} \approx 1.09417 \)[/tex].)
3. Rearrange the Equation:
Substitute [tex]\( e^{0.09} \)[/tex] back into the equation:
[tex]\[ 191.5 = P \times 1.09417 \][/tex]
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{191.5}{1.09417} \][/tex]
5. Calculate [tex]\( P \)[/tex]:
The division results in:
[tex]\[ P \approx 175.0178 \][/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is 175. So, the correct answer is:
B. 175
