Answer :
Sure! Let's solve the problem step by step.
We are given the function [tex]\(f(t) = P e^{rt}\)[/tex] and the information that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex]. Our goal is to find the value of [tex]\( P \)[/tex].
Step 1: Write down the function with the given values.
[tex]\[ f(t) = P e^{rt} \][/tex]
Step 2: Substitute [tex]\( t = 5 \)[/tex], [tex]\( r = 0.05 \)[/tex], and [tex]\( f(5) = 288.9 \)[/tex] into the function.
[tex]\[ 288.9 = P e^{0.05 \times 5} \][/tex]
Step 3: Simplify the exponent.
[tex]\[ 0.05 \times 5 = 0.25 \][/tex]
So the equation becomes:
[tex]\[ 288.9 = P e^{0.25} \][/tex]
Step 4: Solve for [tex]\( P \)[/tex].
First, we need to calculate [tex]\( e^{0.25} \)[/tex]:
[tex]\[ e^{0.25} \approx 1.2840 \][/tex]
Now, substitute [tex]\( e^{0.25} \)[/tex] back into the equation:
[tex]\[ 288.9 = P \times 1.2840 \][/tex]
Step 5: Isolate [tex]\( P \)[/tex] by dividing both sides of the equation by 1.2840.
[tex]\[ P = \frac{288.9}{1.2840} \][/tex]
Step 6: Perform the division.
[tex]\[ P \approx 224.9955 \][/tex]
Step 7: Based on the calculated value of [tex]\( P \)[/tex], we look at the provided options:
A. 371
B. 3520
C. 24
D. 225
The closest value to our calculated [tex]\( P \approx 224.9955 \)[/tex] is option D.
Therefore, the approximate value of [tex]\( P \)[/tex] is:
[tex]\[ \boxed{225} \][/tex]
We are given the function [tex]\(f(t) = P e^{rt}\)[/tex] and the information that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex]. Our goal is to find the value of [tex]\( P \)[/tex].
Step 1: Write down the function with the given values.
[tex]\[ f(t) = P e^{rt} \][/tex]
Step 2: Substitute [tex]\( t = 5 \)[/tex], [tex]\( r = 0.05 \)[/tex], and [tex]\( f(5) = 288.9 \)[/tex] into the function.
[tex]\[ 288.9 = P e^{0.05 \times 5} \][/tex]
Step 3: Simplify the exponent.
[tex]\[ 0.05 \times 5 = 0.25 \][/tex]
So the equation becomes:
[tex]\[ 288.9 = P e^{0.25} \][/tex]
Step 4: Solve for [tex]\( P \)[/tex].
First, we need to calculate [tex]\( e^{0.25} \)[/tex]:
[tex]\[ e^{0.25} \approx 1.2840 \][/tex]
Now, substitute [tex]\( e^{0.25} \)[/tex] back into the equation:
[tex]\[ 288.9 = P \times 1.2840 \][/tex]
Step 5: Isolate [tex]\( P \)[/tex] by dividing both sides of the equation by 1.2840.
[tex]\[ P = \frac{288.9}{1.2840} \][/tex]
Step 6: Perform the division.
[tex]\[ P \approx 224.9955 \][/tex]
Step 7: Based on the calculated value of [tex]\( P \)[/tex], we look at the provided options:
A. 371
B. 3520
C. 24
D. 225
The closest value to our calculated [tex]\( P \approx 224.9955 \)[/tex] is option D.
Therefore, the approximate value of [tex]\( P \)[/tex] is:
[tex]\[ \boxed{225} \][/tex]
