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Answer :
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], follow these steps:
1. Substitute the known values into the equation [tex]\( f(t) = P e^{rt} \)[/tex]. We have:
[tex]\[
288.9 = P e^{0.05 \times 5}
\][/tex]
2. Calculate the exponent in the expression:
[tex]\[
0.05 \times 5 = 0.25
\][/tex]
3. Substitute this exponent into the exponential function to find [tex]\( e^{0.25} \)[/tex]. This value is approximately:
[tex]\[
e^{0.25} \approx 1.284
\][/tex]
4. Substitute [tex]\( e^{0.25} \)[/tex] back into the equation:
[tex]\[
288.9 = P \times 1.284
\][/tex]
5. Solve for [tex]\( P \)[/tex] by dividing both sides of the equation by 1.284:
[tex]\[
P = \frac{288.9}{1.284}
\][/tex]
6. Calculate the approximate value of [tex]\( P \)[/tex]:
[tex]\[
P \approx 225
\][/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is 225.
Therefore, the correct answer is B. 225.
1. Substitute the known values into the equation [tex]\( f(t) = P e^{rt} \)[/tex]. We have:
[tex]\[
288.9 = P e^{0.05 \times 5}
\][/tex]
2. Calculate the exponent in the expression:
[tex]\[
0.05 \times 5 = 0.25
\][/tex]
3. Substitute this exponent into the exponential function to find [tex]\( e^{0.25} \)[/tex]. This value is approximately:
[tex]\[
e^{0.25} \approx 1.284
\][/tex]
4. Substitute [tex]\( e^{0.25} \)[/tex] back into the equation:
[tex]\[
288.9 = P \times 1.284
\][/tex]
5. Solve for [tex]\( P \)[/tex] by dividing both sides of the equation by 1.284:
[tex]\[
P = \frac{288.9}{1.284}
\][/tex]
6. Calculate the approximate value of [tex]\( P \)[/tex]:
[tex]\[
P \approx 225
\][/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is 225.
Therefore, the correct answer is B. 225.
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