Answer :
To find the approximate value of [tex]\( P \)[/tex], let's break down the problem step-by-step:
1. Understand the Function: We have the function [tex]\( r(t) - P \cdot e^t \)[/tex], where [tex]\( r = 0.03 \)[/tex], and we know that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( t = 3 \)[/tex].
2. Substitute the Known Values: Substitute [tex]\( r \)[/tex], [tex]\( t \)[/tex], and [tex]\( f(t) \)[/tex] into the equation:
[tex]\[
0.03 \cdot 3 - P \cdot e^3 = 191.5
\][/tex]
3. Simplify the Equation: Calculate [tex]\( r(t) \)[/tex]:
[tex]\[
r(t) = 0.03 \cdot 3 = 0.09
\][/tex]
So the equation becomes:
[tex]\[
0.09 - P \cdot e^3 = 191.5
\][/tex]
4. Solve for [tex]\( P \)[/tex]: Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P \cdot e^3 = 0.09 - 191.5
\][/tex]
[tex]\[
P \cdot e^3 = -191.41
\][/tex]
[tex]\[
P = \frac{-191.41}{e^3}
\][/tex]
5. Approximate the Value of [tex]\( P \)[/tex]: Calculate [tex]\( e^3 \)[/tex] (approximately 20.085) and solve for [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{-191.41}{20.085} \approx 9.53
\][/tex]
The correct choice for the approximate value of [tex]\( P \)[/tex] is not explicitly listed since none of the options A (78), B (471), C (175), or D (210) match the calculation of approximately 9.53. It appears there might be a misunderstanding with the function setup or options provided.
1. Understand the Function: We have the function [tex]\( r(t) - P \cdot e^t \)[/tex], where [tex]\( r = 0.03 \)[/tex], and we know that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( t = 3 \)[/tex].
2. Substitute the Known Values: Substitute [tex]\( r \)[/tex], [tex]\( t \)[/tex], and [tex]\( f(t) \)[/tex] into the equation:
[tex]\[
0.03 \cdot 3 - P \cdot e^3 = 191.5
\][/tex]
3. Simplify the Equation: Calculate [tex]\( r(t) \)[/tex]:
[tex]\[
r(t) = 0.03 \cdot 3 = 0.09
\][/tex]
So the equation becomes:
[tex]\[
0.09 - P \cdot e^3 = 191.5
\][/tex]
4. Solve for [tex]\( P \)[/tex]: Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P \cdot e^3 = 0.09 - 191.5
\][/tex]
[tex]\[
P \cdot e^3 = -191.41
\][/tex]
[tex]\[
P = \frac{-191.41}{e^3}
\][/tex]
5. Approximate the Value of [tex]\( P \)[/tex]: Calculate [tex]\( e^3 \)[/tex] (approximately 20.085) and solve for [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{-191.41}{20.085} \approx 9.53
\][/tex]
The correct choice for the approximate value of [tex]\( P \)[/tex] is not explicitly listed since none of the options A (78), B (471), C (175), or D (210) match the calculation of approximately 9.53. It appears there might be a misunderstanding with the function setup or options provided.
