Answer :
To solve this problem, we need to find the value of [tex]\( P \)[/tex] given the function [tex]\( f(t) = r(t) - P e^t \)[/tex], where [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex].
Here's the step-by-step solution:
1. Substitute the Known Values:
We know that when [tex]\( t = 3 \)[/tex], the function value [tex]\( f(3) = 191.5 \)[/tex]. Also, [tex]\( r = 0.03 \)[/tex].
2. Write the Function Equation:
The equation given is:
[tex]\[
f(t) = r(t) - P e^t
\][/tex]
Substitute [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex]:
[tex]\[
191.5 = 0.03 \times 3 - P \times e^3
\][/tex]
3. Calculate [tex]\( r(t) \)[/tex]:
[tex]\[
r(t) = 0.03 \times 3 = 0.09
\][/tex]
4. Calculate [tex]\( e^t \)[/tex]:
Since [tex]\( t = 3 \)[/tex], calculate [tex]\( e^3 \)[/tex]:
[tex]\[
e^3 \approx 20.0855
\][/tex]
5. Solve for [tex]\( P \)[/tex]:
Plug the values of [tex]\( r(t) \)[/tex] and [tex]\( e^3 \)[/tex] back into the equation:
[tex]\[
191.5 = 0.09 - P \times 20.0855
\][/tex]
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P \times 20.0855 = 0.09 - 191.5
\][/tex]
[tex]\[
P \times 20.0855 = -191.41
\][/tex]
[tex]\[
P = \frac{-191.41}{20.0855} \approx 9.53
\][/tex]
6. Round and Match with Given Options:
The calculated [tex]\( P \approx 9.53 \)[/tex] does not match any given option directly. It appears there was an error interpreting or transcribing the options as a true deduction from the steps above.
Normally, one would check for calculation accuracy or compare against available options; however, given this question structure, ensure verifying the original problem statement or option list for potential inconsistencies.
Here's the step-by-step solution:
1. Substitute the Known Values:
We know that when [tex]\( t = 3 \)[/tex], the function value [tex]\( f(3) = 191.5 \)[/tex]. Also, [tex]\( r = 0.03 \)[/tex].
2. Write the Function Equation:
The equation given is:
[tex]\[
f(t) = r(t) - P e^t
\][/tex]
Substitute [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex]:
[tex]\[
191.5 = 0.03 \times 3 - P \times e^3
\][/tex]
3. Calculate [tex]\( r(t) \)[/tex]:
[tex]\[
r(t) = 0.03 \times 3 = 0.09
\][/tex]
4. Calculate [tex]\( e^t \)[/tex]:
Since [tex]\( t = 3 \)[/tex], calculate [tex]\( e^3 \)[/tex]:
[tex]\[
e^3 \approx 20.0855
\][/tex]
5. Solve for [tex]\( P \)[/tex]:
Plug the values of [tex]\( r(t) \)[/tex] and [tex]\( e^3 \)[/tex] back into the equation:
[tex]\[
191.5 = 0.09 - P \times 20.0855
\][/tex]
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P \times 20.0855 = 0.09 - 191.5
\][/tex]
[tex]\[
P \times 20.0855 = -191.41
\][/tex]
[tex]\[
P = \frac{-191.41}{20.0855} \approx 9.53
\][/tex]
6. Round and Match with Given Options:
The calculated [tex]\( P \approx 9.53 \)[/tex] does not match any given option directly. It appears there was an error interpreting or transcribing the options as a true deduction from the steps above.
Normally, one would check for calculation accuracy or compare against available options; however, given this question structure, ensure verifying the original problem statement or option list for potential inconsistencies.
