Answer :
Sure, let's solve the problem step-by-step.
We are given a function [tex]\( f(t) = P e^{rt} \)[/tex] and specific values:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
We need to find the approximate value of [tex]\( P \)[/tex].
### Step-by-Step Solution
1. Substitute the known values into the function:
[tex]\( f(5) = P e^{0.05 \cdot 5} \)[/tex]
2. We know [tex]\( f(5) = 288.9 \)[/tex], so the equation becomes:
[tex]\( 288.9 = P e^{0.25} \)[/tex]
3. Calculate [tex]\( e^{0.25} \)[/tex]:
The value of [tex]\( e^{0.25} \)[/tex] is approximately 1.284.
Thus, the equation is:
[tex]\( 288.9 = P \cdot 1.284 \)[/tex]
4. Solve for [tex]\( P \)[/tex]:
Divide both sides by [tex]\( 1.284 \)[/tex]:
[tex]\( P = \frac{288.9}{1.284} \)[/tex]
[tex]\( P \approx 224.995546229 \)[/tex]
5. Identify the closest value from the given options:
The given options are:
- A. 371
- B. 3520
- C. 24
- D. 225
The value we calculated, approximately 225, is closest to option D.
### Answer:
The approximate value of [tex]\( P \)[/tex] is 225.
Option D is the correct answer.
We are given a function [tex]\( f(t) = P e^{rt} \)[/tex] and specific values:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
We need to find the approximate value of [tex]\( P \)[/tex].
### Step-by-Step Solution
1. Substitute the known values into the function:
[tex]\( f(5) = P e^{0.05 \cdot 5} \)[/tex]
2. We know [tex]\( f(5) = 288.9 \)[/tex], so the equation becomes:
[tex]\( 288.9 = P e^{0.25} \)[/tex]
3. Calculate [tex]\( e^{0.25} \)[/tex]:
The value of [tex]\( e^{0.25} \)[/tex] is approximately 1.284.
Thus, the equation is:
[tex]\( 288.9 = P \cdot 1.284 \)[/tex]
4. Solve for [tex]\( P \)[/tex]:
Divide both sides by [tex]\( 1.284 \)[/tex]:
[tex]\( P = \frac{288.9}{1.284} \)[/tex]
[tex]\( P \approx 224.995546229 \)[/tex]
5. Identify the closest value from the given options:
The given options are:
- A. 371
- B. 3520
- C. 24
- D. 225
The value we calculated, approximately 225, is closest to option D.
### Answer:
The approximate value of [tex]\( P \)[/tex] is 225.
Option D is the correct answer.
