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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]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex].
Here are the step-by-step instructions:
1. Understand the Function: The function provided is [tex]\( f(t) = P e^{rt} \)[/tex]. This is an exponential function where [tex]\( P \)[/tex] is the initial value, and [tex]\( r \)[/tex] is the rate of growth.
2. Substitute Known Values: We know:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]
Substitute these values into the function:
[tex]\[
288.9 = P \cdot e^{0.05 \times 5}
\][/tex]
3. Calculate [tex]\( e^{0.25} \)[/tex]: The exponent is calculated by multiplying the rate by the time:
[tex]\[
rt = 0.05 \times 5 = 0.25
\][/tex]
Evaluate the expression [tex]\( e^{0.25} \)[/tex]. Calculating this gives approximately:
[tex]\[
e^{0.25} \approx 1.284
\][/tex]
4. Solve for [tex]\( P \)[/tex]: Now solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( e^{0.25} \)[/tex]:
[tex]\[
P = \frac{288.9}{1.284}
\][/tex]
5. Calculate [tex]\( P \)[/tex]: Carry out the division:
[tex]\[
P \approx \frac{288.9}{1.284} \approx 225
\][/tex]
6. Select the Closest Answer: After calculating, the approximate value of [tex]\( P \)[/tex] is 225, which matches option B.
Therefore, the answer is B. 225.
Here are the step-by-step instructions:
1. Understand the Function: The function provided is [tex]\( f(t) = P e^{rt} \)[/tex]. This is an exponential function where [tex]\( P \)[/tex] is the initial value, and [tex]\( r \)[/tex] is the rate of growth.
2. Substitute Known Values: We know:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]
Substitute these values into the function:
[tex]\[
288.9 = P \cdot e^{0.05 \times 5}
\][/tex]
3. Calculate [tex]\( e^{0.25} \)[/tex]: The exponent is calculated by multiplying the rate by the time:
[tex]\[
rt = 0.05 \times 5 = 0.25
\][/tex]
Evaluate the expression [tex]\( e^{0.25} \)[/tex]. Calculating this gives approximately:
[tex]\[
e^{0.25} \approx 1.284
\][/tex]
4. Solve for [tex]\( P \)[/tex]: Now solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( e^{0.25} \)[/tex]:
[tex]\[
P = \frac{288.9}{1.284}
\][/tex]
5. Calculate [tex]\( P \)[/tex]: Carry out the division:
[tex]\[
P \approx \frac{288.9}{1.284} \approx 225
\][/tex]
6. Select the Closest Answer: After calculating, the approximate value of [tex]\( P \)[/tex] is 225, which matches option B.
Therefore, the answer is B. 225.
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