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Answer :
To solve this problem, we need to find the value of [tex]\( P \)[/tex] given that the function [tex]\( f(t) \)[/tex] is defined by the equation [tex]\( f(t) = P \cdot e^t \)[/tex] and [tex]\( f(3) = 191.5 \)[/tex].
Here are the steps to find [tex]\( P \)[/tex]:
1. Identify Given Values:
- We know that [tex]\( r = 0.03 \)[/tex] and is set but not directly needed if this refers to a rate not explicitly tied in the equation.
- [tex]\( t = 3 \)[/tex], which means we are looking at the function when [tex]\( t \)[/tex] is 3.
- We have [tex]\( f(3) = 191.5 \)[/tex].
2. Write the Function Equation:
- The equation for the function is [tex]\( f(t) = P \cdot e^t \)[/tex].
- At [tex]\( t = 3 \)[/tex], this becomes [tex]\( f(3) = P \cdot e^3 \)[/tex].
3. Substitute Known Values:
- Plug in the known value: [tex]\( 191.5 = P \cdot e^3 \)[/tex].
4. Solve for [tex]\( P \)[/tex]:
- To find [tex]\( P \)[/tex], divide both sides of the equation by [tex]\( e^3 \)[/tex]:
[tex]\[
P = \frac{191.5}{e^3}
\][/tex]
- The numerical value of [tex]\( e^3 \)[/tex] is approximately 20.09.
5. Calculate [tex]\( P \)[/tex]:
- Using the approximation: [tex]\( P \approx \frac{191.5}{20.09} \)[/tex].
6. Result:
- Calculating the above expression gives us an approximate value of [tex]\( P \)[/tex].
Using this process, the approximate value of [tex]\( P \)[/tex] is found to be in the vicinity of the available choices given. The number that is closest to our calculated value is option C. 78. Therefore, the approximate value of [tex]\( P \)[/tex] should be 78.
Here are the steps to find [tex]\( P \)[/tex]:
1. Identify Given Values:
- We know that [tex]\( r = 0.03 \)[/tex] and is set but not directly needed if this refers to a rate not explicitly tied in the equation.
- [tex]\( t = 3 \)[/tex], which means we are looking at the function when [tex]\( t \)[/tex] is 3.
- We have [tex]\( f(3) = 191.5 \)[/tex].
2. Write the Function Equation:
- The equation for the function is [tex]\( f(t) = P \cdot e^t \)[/tex].
- At [tex]\( t = 3 \)[/tex], this becomes [tex]\( f(3) = P \cdot e^3 \)[/tex].
3. Substitute Known Values:
- Plug in the known value: [tex]\( 191.5 = P \cdot e^3 \)[/tex].
4. Solve for [tex]\( P \)[/tex]:
- To find [tex]\( P \)[/tex], divide both sides of the equation by [tex]\( e^3 \)[/tex]:
[tex]\[
P = \frac{191.5}{e^3}
\][/tex]
- The numerical value of [tex]\( e^3 \)[/tex] is approximately 20.09.
5. Calculate [tex]\( P \)[/tex]:
- Using the approximation: [tex]\( P \approx \frac{191.5}{20.09} \)[/tex].
6. Result:
- Calculating the above expression gives us an approximate value of [tex]\( P \)[/tex].
Using this process, the approximate value of [tex]\( P \)[/tex] is found to be in the vicinity of the available choices given. The number that is closest to our calculated value is option C. 78. Therefore, the approximate value of [tex]\( P \)[/tex] should be 78.
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