Answer :
To solve the problem, we need to find the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex] given that [tex]\( f(5) = 288.9 \)[/tex] when [tex]\( r = 0.05 \)[/tex].
Here's how you can work it out step-by-step:
1. Understand the given equation:
[tex]\[ f(t) = P e^{rt} \][/tex]
We know:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]
Substituting these values into the function gives:
[tex]\[ 288.9 = P e^{0.05 \times 5} \][/tex]
2. Calculate the exponent:
[tex]\[ r \times t = 0.05 \times 5 = 0.25 \][/tex]
3. Calculate [tex]\( e^{0.25} \)[/tex]:
The approximate value of [tex]\( e^{0.25} \)[/tex] is about 1.284.
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[ 288.9 = P \times 1.284 \][/tex]
Divide both sides by 1.284:
[tex]\[ P = \frac{288.9}{1.284} \][/tex]
Calculating this gives:
[tex]\[ P \approx 225 \][/tex]
5. Conclusion:
Therefore, the approximate value of [tex]\( P \)[/tex] is 225.
So the correct answer choice is C. 225.
Here's how you can work it out step-by-step:
1. Understand the given equation:
[tex]\[ f(t) = P e^{rt} \][/tex]
We know:
- [tex]\( f(5) = 288.9 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( t = 5 \)[/tex]
Substituting these values into the function gives:
[tex]\[ 288.9 = P e^{0.05 \times 5} \][/tex]
2. Calculate the exponent:
[tex]\[ r \times t = 0.05 \times 5 = 0.25 \][/tex]
3. Calculate [tex]\( e^{0.25} \)[/tex]:
The approximate value of [tex]\( e^{0.25} \)[/tex] is about 1.284.
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[ 288.9 = P \times 1.284 \][/tex]
Divide both sides by 1.284:
[tex]\[ P = \frac{288.9}{1.284} \][/tex]
Calculating this gives:
[tex]\[ P \approx 225 \][/tex]
5. Conclusion:
Therefore, the approximate value of [tex]\( P \)[/tex] is 225.
So the correct answer choice is C. 225.
