Answer :
To find the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^t \)[/tex] given that [tex]\( f(3) = 191.5 \)[/tex], we can follow these steps:
1. Understand the Problem: We are given the function [tex]\( f(t) = P e^t \)[/tex] and know that when [tex]\( t = 3 \)[/tex], the value of the function is 191.5. Our task is to find the value of [tex]\( P \)[/tex].
2. Set Up the Equation: When [tex]\( t = 3 \)[/tex], the function becomes:
[tex]\[
f(3) = P e^3 = 191.5
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
- To isolate [tex]\( P \)[/tex], divide both sides of the equation by [tex]\( e^3 \)[/tex]:
[tex]\[
P = \frac{191.5}{e^3}
\][/tex]
4. Calculate the Value:
- Use the approximate value of [tex]\( e \)[/tex] (Euler's number), which is approximately 2.71828. Raise this to the power of 3:
[tex]\[
e^3 \approx 2.71828^3 \approx 20.0855
\][/tex]
- Now, divide 191.5 by this value:
[tex]\[
P \approx \frac{191.5}{20.0855} \approx 9.53
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 9.53. Since none of the choices given (A: 210, B: 175, C: 471, D: 78) are close to 9.53, it seems there was no suitable answer option provided in the list.
1. Understand the Problem: We are given the function [tex]\( f(t) = P e^t \)[/tex] and know that when [tex]\( t = 3 \)[/tex], the value of the function is 191.5. Our task is to find the value of [tex]\( P \)[/tex].
2. Set Up the Equation: When [tex]\( t = 3 \)[/tex], the function becomes:
[tex]\[
f(3) = P e^3 = 191.5
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
- To isolate [tex]\( P \)[/tex], divide both sides of the equation by [tex]\( e^3 \)[/tex]:
[tex]\[
P = \frac{191.5}{e^3}
\][/tex]
4. Calculate the Value:
- Use the approximate value of [tex]\( e \)[/tex] (Euler's number), which is approximately 2.71828. Raise this to the power of 3:
[tex]\[
e^3 \approx 2.71828^3 \approx 20.0855
\][/tex]
- Now, divide 191.5 by this value:
[tex]\[
P \approx \frac{191.5}{20.0855} \approx 9.53
\][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 9.53. Since none of the choices given (A: 210, B: 175, C: 471, D: 78) are close to 9.53, it seems there was no suitable answer option provided in the list.
