Answer :
Certainly! Let's solve for [tex]\( P \)[/tex] given the function [tex]\( f(t) = P e^{rt} \)[/tex] with the values [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex].
Here's the detailed step-by-step solution:
1. Write down the given information:
- [tex]\( f(t) = P e^{rt} \)[/tex]
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- [tex]\( t = 3 \)[/tex]
2. Substitute the known values into the function:
[tex]\[
191.5 = P e^{0.03 \times 3}
\][/tex]
3. Simplify the exponent:
[tex]\[
0.03 \times 3 = 0.09
\][/tex]
4. Rewrite the equation with this value:
[tex]\[
191.5 = P e^{0.09}
\][/tex]
5. Solve for [tex]\( P \)[/tex] by isolating it on one side of the equation:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
6. Calculate the value of [tex]\( e^{0.09} \)[/tex]:
Using [tex]\( e \approx 2.718 \)[/tex], we find [tex]\( e^{0.09} \approx 1.094174 \)[/tex] (Though this value can be found using a calculator or other computational means).
7. Divide 191.5 by [tex]\( e^{0.09} \)[/tex]:
[tex]\[
P \approx \frac{191.5}{1.094174} \approx 175.02
\][/tex]
The value of [tex]\( P \)[/tex] is approximately 175.
Therefore, the answer is B. 175.
Here's the detailed step-by-step solution:
1. Write down the given information:
- [tex]\( f(t) = P e^{rt} \)[/tex]
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- [tex]\( t = 3 \)[/tex]
2. Substitute the known values into the function:
[tex]\[
191.5 = P e^{0.03 \times 3}
\][/tex]
3. Simplify the exponent:
[tex]\[
0.03 \times 3 = 0.09
\][/tex]
4. Rewrite the equation with this value:
[tex]\[
191.5 = P e^{0.09}
\][/tex]
5. Solve for [tex]\( P \)[/tex] by isolating it on one side of the equation:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
6. Calculate the value of [tex]\( e^{0.09} \)[/tex]:
Using [tex]\( e \approx 2.718 \)[/tex], we find [tex]\( e^{0.09} \approx 1.094174 \)[/tex] (Though this value can be found using a calculator or other computational means).
7. Divide 191.5 by [tex]\( e^{0.09} \)[/tex]:
[tex]\[
P \approx \frac{191.5}{1.094174} \approx 175.02
\][/tex]
The value of [tex]\( P \)[/tex] is approximately 175.
Therefore, the answer is B. 175.
