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Answer :
To find the approximate value of [tex]\( P \)[/tex] given the function [tex]\( f(t) = P e^{rt} \)[/tex], where [tex]\( f(5) = 288.9 \)[/tex], [tex]\( r = 0.05 \)[/tex], and [tex]\( t = 5 \)[/tex], follow these steps:
1. Understand the equation:
The equation given is [tex]\( f(t) = P e^{rt} \)[/tex]. This is an exponential function where [tex]\( P \)[/tex] is an initial amount, [tex]\( r \)[/tex] is the growth rate, and [tex]\( t \)[/tex] is the time.
2. Substitute the known values:
We know that when [tex]\( t = 5 \)[/tex], [tex]\( f(5) = 288.9 \)[/tex]. So, we substitute these values into the equation:
[tex]\[
288.9 = P e^{0.05 \times 5}
\][/tex]
3. Calculate [tex]\( e^{0.05 \times 5} \)[/tex]:
First, calculate the product of [tex]\( r \)[/tex] and [tex]\( t \)[/tex]:
[tex]\[
0.05 \times 5 = 0.25
\][/tex]
Then, calculate [tex]\( e^{0.25} \)[/tex] which is a constant approximately equal to 1.2840.
4. Solve for [tex]\( P \)[/tex]:
Substitute the value of [tex]\( e^{0.25} \)[/tex] into the equation:
[tex]\[
288.9 = P \cdot 1.2840
\][/tex]
To find [tex]\( P \)[/tex], divide both sides by 1.2840:
[tex]\[
P = \frac{288.9}{1.2840} \approx 224.9955
\][/tex]
5. Choose the closest answer:
From the options given:
- A: 225
- B: 24
- C: 371
- D: 3520
The answer closest to 224.9955 is 225.
So, the approximate value of [tex]\( P \)[/tex] is 225. Therefore, the correct choice is A. 225.
1. Understand the equation:
The equation given is [tex]\( f(t) = P e^{rt} \)[/tex]. This is an exponential function where [tex]\( P \)[/tex] is an initial amount, [tex]\( r \)[/tex] is the growth rate, and [tex]\( t \)[/tex] is the time.
2. Substitute the known values:
We know that when [tex]\( t = 5 \)[/tex], [tex]\( f(5) = 288.9 \)[/tex]. So, we substitute these values into the equation:
[tex]\[
288.9 = P e^{0.05 \times 5}
\][/tex]
3. Calculate [tex]\( e^{0.05 \times 5} \)[/tex]:
First, calculate the product of [tex]\( r \)[/tex] and [tex]\( t \)[/tex]:
[tex]\[
0.05 \times 5 = 0.25
\][/tex]
Then, calculate [tex]\( e^{0.25} \)[/tex] which is a constant approximately equal to 1.2840.
4. Solve for [tex]\( P \)[/tex]:
Substitute the value of [tex]\( e^{0.25} \)[/tex] into the equation:
[tex]\[
288.9 = P \cdot 1.2840
\][/tex]
To find [tex]\( P \)[/tex], divide both sides by 1.2840:
[tex]\[
P = \frac{288.9}{1.2840} \approx 224.9955
\][/tex]
5. Choose the closest answer:
From the options given:
- A: 225
- B: 24
- C: 371
- D: 3520
The answer closest to 224.9955 is 225.
So, the approximate value of [tex]\( P \)[/tex] is 225. Therefore, the correct choice is A. 225.
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