Answer :
To find the approximate value of [tex]\( P \)[/tex] when [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex] for the function defined as [tex]\( f(t) = r \cdot t - P \cdot e^t \)[/tex], we can proceed step-by-step:
1. Understand the Equation:
The function is given as [tex]\( f(t) = r \cdot t - P \cdot e^t \)[/tex].
2. Plug in the Given Values:
- We know that when [tex]\( t = 3 \)[/tex], the function value, [tex]\( f(3) \)[/tex], is 191.5.
- [tex]\( r = 0.03 \)[/tex].
3. Set Up the Equation for [tex]\( f(3) \)[/tex]:
Substitute the known values into the equation:
[tex]\[
f(3) = r \cdot 3 - P \cdot e^3
\][/tex]
[tex]\[
191.5 = (0.03 \cdot 3) - P \cdot e^3
\][/tex]
4. Calculate [tex]\( r \cdot 3 \)[/tex]:
- [tex]\( 0.03 \times 3 = 0.09 \)[/tex].
5. Calculate [tex]\( e^3 \)[/tex]:
- The approximate value of [tex]\( e^3 \)[/tex] is 20.086.
6. Rearrange and Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
191.5 = 0.09 - P \cdot 20.086
\][/tex]
[tex]\[
P \cdot 20.086 = 0.09 - 191.5
\][/tex]
[tex]\[
P = \frac{0.09 - 191.5}{-20.086}
\][/tex]
7. Calculate [tex]\( P \)[/tex]:
- Find the value of the numerator: [tex]\( 0.09 - 191.5 \approx -191.41 \)[/tex].
- Now, divide by [tex]\(-20.086\)[/tex]:
[tex]\[
P = \frac{-191.41}{-20.086} \approx 9.53
\][/tex]
The approximate value of [tex]\( P \)[/tex] is close to 9.53, which is not in the options provided (210, 78, 175, 471). It seems there might be some miscommunication in the problem setup or possible errors in options. Based on our calculations, [tex]\( P \approx 9.53 \)[/tex].
1. Understand the Equation:
The function is given as [tex]\( f(t) = r \cdot t - P \cdot e^t \)[/tex].
2. Plug in the Given Values:
- We know that when [tex]\( t = 3 \)[/tex], the function value, [tex]\( f(3) \)[/tex], is 191.5.
- [tex]\( r = 0.03 \)[/tex].
3. Set Up the Equation for [tex]\( f(3) \)[/tex]:
Substitute the known values into the equation:
[tex]\[
f(3) = r \cdot 3 - P \cdot e^3
\][/tex]
[tex]\[
191.5 = (0.03 \cdot 3) - P \cdot e^3
\][/tex]
4. Calculate [tex]\( r \cdot 3 \)[/tex]:
- [tex]\( 0.03 \times 3 = 0.09 \)[/tex].
5. Calculate [tex]\( e^3 \)[/tex]:
- The approximate value of [tex]\( e^3 \)[/tex] is 20.086.
6. Rearrange and Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
191.5 = 0.09 - P \cdot 20.086
\][/tex]
[tex]\[
P \cdot 20.086 = 0.09 - 191.5
\][/tex]
[tex]\[
P = \frac{0.09 - 191.5}{-20.086}
\][/tex]
7. Calculate [tex]\( P \)[/tex]:
- Find the value of the numerator: [tex]\( 0.09 - 191.5 \approx -191.41 \)[/tex].
- Now, divide by [tex]\(-20.086\)[/tex]:
[tex]\[
P = \frac{-191.41}{-20.086} \approx 9.53
\][/tex]
The approximate value of [tex]\( P \)[/tex] is close to 9.53, which is not in the options provided (210, 78, 175, 471). It seems there might be some miscommunication in the problem setup or possible errors in options. Based on our calculations, [tex]\( P \approx 9.53 \)[/tex].
