Answer :
To solve this problem, we need to find the value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex] when given specific values. Here's how we can do that step-by-step:
1. Understand the Given Information:
We have [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].
2. Use the Function Formula:
The given function is [tex]\( f(t) = P e^{rt} \)[/tex].
3. Substitute the Known Values:
We know that [tex]\( t = 3 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( f(3) = 191.5 \)[/tex]. Substitute these into the function:
[tex]\[
191.5 = P \cdot e^{0.03 \times 3}
\][/tex]
4. Calculate the Exponential Part:
Calculate [tex]\( e^{0.03 \times 3} = e^{0.09} \)[/tex].
5. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
6. Calculate the Value of [tex]\( P \)[/tex]:
Evaluate the right side of the equation to find the approximate value of [tex]\( P \)[/tex].
Using these steps, the approximate value of [tex]\( P \)[/tex] is very close to 175. Therefore, among the given options, the correct answer is:
D. 175
1. Understand the Given Information:
We have [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].
2. Use the Function Formula:
The given function is [tex]\( f(t) = P e^{rt} \)[/tex].
3. Substitute the Known Values:
We know that [tex]\( t = 3 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( f(3) = 191.5 \)[/tex]. Substitute these into the function:
[tex]\[
191.5 = P \cdot e^{0.03 \times 3}
\][/tex]
4. Calculate the Exponential Part:
Calculate [tex]\( e^{0.03 \times 3} = e^{0.09} \)[/tex].
5. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
6. Calculate the Value of [tex]\( P \)[/tex]:
Evaluate the right side of the equation to find the approximate value of [tex]\( P \)[/tex].
Using these steps, the approximate value of [tex]\( P \)[/tex] is very close to 175. Therefore, among the given options, the correct answer is:
D. 175
