Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex], we can use the given information:
1. We know that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex].
2. The function is given by [tex]\( f(t) = P e^{rt} \)[/tex].
Given these values, let's calculate the approximate value of [tex]\( P \)[/tex] step-by-step:
### Step 1: Set Up the Equation
We have:
[tex]\[ f(5) = P e^{0.05 \times 5} \][/tex]
We know:
[tex]\[ f(5) = 288.9 \][/tex]
### Step 2: Solve for [tex]\( P \)[/tex]
Substitute the known value of [tex]\( f(5) \)[/tex] into the equation:
[tex]\[ 288.9 = P e^{0.05 \times 5} \][/tex]
Calculate [tex]\( e^{0.05 \times 5} \)[/tex]:
[tex]\[ e^{0.25} \approx 1.284 \][/tex]
Now, solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{288.9}{1.284} \][/tex]
Perform the division to find [tex]\( P \)[/tex]:
[tex]\[ P \approx 224.995 \][/tex]
### Conclusion
Therefore, the approximate value of [tex]\( P \)[/tex] is about 225. Given the options:
A. 3520
B. 371
C. 24
D. 225
The correct answer is D. 225.
1. We know that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex].
2. The function is given by [tex]\( f(t) = P e^{rt} \)[/tex].
Given these values, let's calculate the approximate value of [tex]\( P \)[/tex] step-by-step:
### Step 1: Set Up the Equation
We have:
[tex]\[ f(5) = P e^{0.05 \times 5} \][/tex]
We know:
[tex]\[ f(5) = 288.9 \][/tex]
### Step 2: Solve for [tex]\( P \)[/tex]
Substitute the known value of [tex]\( f(5) \)[/tex] into the equation:
[tex]\[ 288.9 = P e^{0.05 \times 5} \][/tex]
Calculate [tex]\( e^{0.05 \times 5} \)[/tex]:
[tex]\[ e^{0.25} \approx 1.284 \][/tex]
Now, solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{288.9}{1.284} \][/tex]
Perform the division to find [tex]\( P \)[/tex]:
[tex]\[ P \approx 224.995 \][/tex]
### Conclusion
Therefore, the approximate value of [tex]\( P \)[/tex] is about 225. Given the options:
A. 3520
B. 371
C. 24
D. 225
The correct answer is D. 225.
