Answer :
Sure, I'll help you solve this problem step-by-step.
We are given the function:
[tex]\[ f(t) = P e^{rt} \][/tex]
where:
- [tex]\( f(t) = 288.9 \)[/tex] when [tex]\( t = 5 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
We need to find the value of [tex]\( P \)[/tex].
### Step-by-Step Solution:
1. Plug in the given values into the function:
[tex]\[ f(5) = P e^{0.05 \cdot 5} \][/tex]
2. Substitute [tex]\( f(5) \)[/tex] with 288.9:
[tex]\[ 288.9 = P \cdot e^{0.25} \][/tex]
3. Calculate [tex]\( e^{0.25} \)[/tex]:
Using a calculator or referencing the numerical result, we have:
[tex]\[ e^{0.25} \approx 1.2840254166877414 \][/tex]
4. Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{288.9}{1.2840254166877414} \][/tex]
5. Calculate the value of [tex]\( P \)[/tex]:
[tex]\[ P \approx 224.99554622932885 \][/tex]
The approximate value of [tex]\( P \)[/tex] is very close to 225.
### Final Answer:
[tex]\[ \boxed{225} \][/tex]
So, the correct answer is A. 225.
We are given the function:
[tex]\[ f(t) = P e^{rt} \][/tex]
where:
- [tex]\( f(t) = 288.9 \)[/tex] when [tex]\( t = 5 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
We need to find the value of [tex]\( P \)[/tex].
### Step-by-Step Solution:
1. Plug in the given values into the function:
[tex]\[ f(5) = P e^{0.05 \cdot 5} \][/tex]
2. Substitute [tex]\( f(5) \)[/tex] with 288.9:
[tex]\[ 288.9 = P \cdot e^{0.25} \][/tex]
3. Calculate [tex]\( e^{0.25} \)[/tex]:
Using a calculator or referencing the numerical result, we have:
[tex]\[ e^{0.25} \approx 1.2840254166877414 \][/tex]
4. Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{288.9}{1.2840254166877414} \][/tex]
5. Calculate the value of [tex]\( P \)[/tex]:
[tex]\[ P \approx 224.99554622932885 \][/tex]
The approximate value of [tex]\( P \)[/tex] is very close to 225.
### Final Answer:
[tex]\[ \boxed{225} \][/tex]
So, the correct answer is A. 225.
