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Answer :
To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex] where [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], we can follow these steps:
1. Understand the problem setup: The function is given by [tex]\( f(t) = P e^{rt} \)[/tex]. We're told that when [tex]\( t = 5 \)[/tex], the function [tex]\( f(5) \)[/tex] equals 288.9. We need to find [tex]\( P \)[/tex].
2. Set up the equation with given values: We have the equation [tex]\( f(5) = P e^{0.05 \times 5} = 288.9 \)[/tex].
3. Simplify the expression inside the exponential: Calculate [tex]\( 0.05 \times 5 = 0.25 \)[/tex].
4. Substitute and solve for [tex]\( P \)[/tex]:
[tex]\[
P \cdot e^{0.25} = 288.9
\][/tex]
5. Estimate the value of [tex]\( e^{0.25} \)[/tex]: The number [tex]\( e^{0.25} \)[/tex] is approximately equal to 1.284.
6. Solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{1.284}
\][/tex]
7. Calculate [tex]\( P \)[/tex]:
[tex]\[
P \approx 224.996
\][/tex]
Thus, the value of [tex]\( P \)[/tex] is approximately 225. Therefore, the correct answer is:
D. 225
1. Understand the problem setup: The function is given by [tex]\( f(t) = P e^{rt} \)[/tex]. We're told that when [tex]\( t = 5 \)[/tex], the function [tex]\( f(5) \)[/tex] equals 288.9. We need to find [tex]\( P \)[/tex].
2. Set up the equation with given values: We have the equation [tex]\( f(5) = P e^{0.05 \times 5} = 288.9 \)[/tex].
3. Simplify the expression inside the exponential: Calculate [tex]\( 0.05 \times 5 = 0.25 \)[/tex].
4. Substitute and solve for [tex]\( P \)[/tex]:
[tex]\[
P \cdot e^{0.25} = 288.9
\][/tex]
5. Estimate the value of [tex]\( e^{0.25} \)[/tex]: The number [tex]\( e^{0.25} \)[/tex] is approximately equal to 1.284.
6. Solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{1.284}
\][/tex]
7. Calculate [tex]\( P \)[/tex]:
[tex]\[
P \approx 224.996
\][/tex]
Thus, the value of [tex]\( P \)[/tex] is approximately 225. Therefore, the correct answer is:
D. 225
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