Answer :
To solve this problem, we need to find the value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex]. We are given:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
We can use these values to find [tex]\( P \)[/tex] by following these steps:
1. Set up the equation:
The function given is [tex]\( f(t) = P e^{rt} \)[/tex]. We know [tex]\( f(4) = 246.4 \)[/tex], which gives us:
[tex]\[ 246.4 = P \times e^{0.04 \times 4} \][/tex]
2. Simplify the exponent:
First, calculate the exponent [tex]\( rt \)[/tex]:
[tex]\[ rt = 0.04 \times 4 = 0.16 \][/tex]
3. Calculate [tex]\( e^{0.16} \)[/tex]:
We need [tex]\( e^{0.16} \)[/tex] to find [tex]\( P \)[/tex]. This value is approximately 1.1735.
4. Solve for [tex]\( P \)[/tex]:
Now substitute the value of [tex]\( e^{0.16} \)[/tex] back into the equation:
[tex]\[ 246.4 = P \times 1.1735 \][/tex]
To find [tex]\( P \)[/tex], divide both sides by 1.1735:
[tex]\[ P = \frac{246.4}{1.1735} \approx 209.97 \][/tex]
5. Choose the closest answer:
Looking at the options provided:
- A. 1220
- B. 289
- C. 210
- D. 50
The closest value to 209.97 is 210.
Thus, the approximate value of [tex]\( P \)[/tex] is C. 210.
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
We can use these values to find [tex]\( P \)[/tex] by following these steps:
1. Set up the equation:
The function given is [tex]\( f(t) = P e^{rt} \)[/tex]. We know [tex]\( f(4) = 246.4 \)[/tex], which gives us:
[tex]\[ 246.4 = P \times e^{0.04 \times 4} \][/tex]
2. Simplify the exponent:
First, calculate the exponent [tex]\( rt \)[/tex]:
[tex]\[ rt = 0.04 \times 4 = 0.16 \][/tex]
3. Calculate [tex]\( e^{0.16} \)[/tex]:
We need [tex]\( e^{0.16} \)[/tex] to find [tex]\( P \)[/tex]. This value is approximately 1.1735.
4. Solve for [tex]\( P \)[/tex]:
Now substitute the value of [tex]\( e^{0.16} \)[/tex] back into the equation:
[tex]\[ 246.4 = P \times 1.1735 \][/tex]
To find [tex]\( P \)[/tex], divide both sides by 1.1735:
[tex]\[ P = \frac{246.4}{1.1735} \approx 209.97 \][/tex]
5. Choose the closest answer:
Looking at the options provided:
- A. 1220
- B. 289
- C. 210
- D. 50
The closest value to 209.97 is 210.
Thus, the approximate value of [tex]\( P \)[/tex] is C. 210.
