Answer :
To solve this problem, we need to find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex], given that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].
Step 1: Set up the equation.
We are given the function:
[tex]\[ f(t) = P e^{rt} \][/tex]
Substitute the given values into the function:
[tex]\[ f(3) = P e^{0.03 \times 3} \][/tex]
We know that [tex]\( f(3) = 191.5 \)[/tex], so:
[tex]\[ 191.5 = P e^{0.09} \][/tex]
Step 2: Solve for [tex]\( P \)[/tex].
To isolate [tex]\( P \)[/tex], we rearrange the equation:
[tex]\[ P = \frac{191.5}{e^{0.09}} \][/tex]
Step 3: Calculate [tex]\( e^{0.09} \)[/tex].
The exponential term [tex]\( e^{0.09} \)[/tex] is a constant:
[tex]\[ e^{0.09} \approx 1.094174 \][/tex]
Step 4: Calculate [tex]\( P \)[/tex].
Now plug the value of [tex]\( e^{0.09} \)[/tex] into the equation for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{191.5}{1.094174} \][/tex]
This gives:
[tex]\[ P \approx 175.01782 \][/tex]
Conclusion:
The approximate value of [tex]\( P \)[/tex] is 175.
Therefore, the answer is:
D. 175
Step 1: Set up the equation.
We are given the function:
[tex]\[ f(t) = P e^{rt} \][/tex]
Substitute the given values into the function:
[tex]\[ f(3) = P e^{0.03 \times 3} \][/tex]
We know that [tex]\( f(3) = 191.5 \)[/tex], so:
[tex]\[ 191.5 = P e^{0.09} \][/tex]
Step 2: Solve for [tex]\( P \)[/tex].
To isolate [tex]\( P \)[/tex], we rearrange the equation:
[tex]\[ P = \frac{191.5}{e^{0.09}} \][/tex]
Step 3: Calculate [tex]\( e^{0.09} \)[/tex].
The exponential term [tex]\( e^{0.09} \)[/tex] is a constant:
[tex]\[ e^{0.09} \approx 1.094174 \][/tex]
Step 4: Calculate [tex]\( P \)[/tex].
Now plug the value of [tex]\( e^{0.09} \)[/tex] into the equation for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{191.5}{1.094174} \][/tex]
This gives:
[tex]\[ P \approx 175.01782 \][/tex]
Conclusion:
The approximate value of [tex]\( P \)[/tex] is 175.
Therefore, the answer is:
D. 175
