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Answer :
To find the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex] given that [tex]\( f(3) = 191.5 \)[/tex] and [tex]\( r = 0.03 \)[/tex], follow these steps:
1. Understand the Equation: We are given the equation [tex]\( f(t) = P e^{rt} \)[/tex]. In our specific case, when [tex]\( t = 3 \)[/tex], [tex]\( f(3) = 191.5 \)[/tex].
2. Set Up the Equation: We substitute the known values into the function:
[tex]\[
191.5 = P e^{0.03 \times 3}
\][/tex]
3. Calculate the Exponential Part: First, calculate [tex]\( e^{0.03 \times 3} \)[/tex].
4. Solve for [tex]\( P \)[/tex]: Use the equation from step 2:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Compute [tex]\( e^{0.09} \)[/tex]: Calculate this value using a calculator to find that [tex]\( e^{0.09} \)[/tex] is approximately 1.094174.
6. Divide to Find [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{1.094174} \approx 175.02
\][/tex]
7. Choose the Closest Answer: From the provided options, [tex]\( 175.02 \)[/tex] is closest to 175.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{175} \)[/tex].
1. Understand the Equation: We are given the equation [tex]\( f(t) = P e^{rt} \)[/tex]. In our specific case, when [tex]\( t = 3 \)[/tex], [tex]\( f(3) = 191.5 \)[/tex].
2. Set Up the Equation: We substitute the known values into the function:
[tex]\[
191.5 = P e^{0.03 \times 3}
\][/tex]
3. Calculate the Exponential Part: First, calculate [tex]\( e^{0.03 \times 3} \)[/tex].
4. Solve for [tex]\( P \)[/tex]: Use the equation from step 2:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Compute [tex]\( e^{0.09} \)[/tex]: Calculate this value using a calculator to find that [tex]\( e^{0.09} \)[/tex] is approximately 1.094174.
6. Divide to Find [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{1.094174} \approx 175.02
\][/tex]
7. Choose the Closest Answer: From the provided options, [tex]\( 175.02 \)[/tex] is closest to 175.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{175} \)[/tex].
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