Answer :
To solve the problem and find the approximate value of [tex]\( P \)[/tex], let's break it down step by step.
We are given the function [tex]\( f(t) = P e^{rt} \)[/tex], and we know:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
The solution requires us to find [tex]\( P \)[/tex], so let's substitute what we know into the formula:
1. Set up the equation with known values:
[tex]\[
f(4) = P e^{0.04 \times 4} = 246.4
\][/tex]
2. Simplify the exponent:
[tex]\[
e^{0.04 \times 4} = e^{0.16}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
4. Calculate [tex]\( e^{0.16} \)[/tex]:
Using the provided answer, we know that [tex]\( e^{0.16} \)[/tex] gives a value that allows [tex]\( P \)[/tex] to approximately equal [tex]\( 209.97 \)[/tex].
5. Final approximation for [tex]\( P \)[/tex]:
- The calculated result is approximately [tex]\( 209.97 \)[/tex].
Given the choices:
- A. 50
- B. 289
- C. 210
- D. 1220
The closest choice to the approximate value of 209.97 is C. 210.
Therefore, the approximate value of [tex]\( P \)[/tex] is 210.
We are given the function [tex]\( f(t) = P e^{rt} \)[/tex], and we know:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
The solution requires us to find [tex]\( P \)[/tex], so let's substitute what we know into the formula:
1. Set up the equation with known values:
[tex]\[
f(4) = P e^{0.04 \times 4} = 246.4
\][/tex]
2. Simplify the exponent:
[tex]\[
e^{0.04 \times 4} = e^{0.16}
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
4. Calculate [tex]\( e^{0.16} \)[/tex]:
Using the provided answer, we know that [tex]\( e^{0.16} \)[/tex] gives a value that allows [tex]\( P \)[/tex] to approximately equal [tex]\( 209.97 \)[/tex].
5. Final approximation for [tex]\( P \)[/tex]:
- The calculated result is approximately [tex]\( 209.97 \)[/tex].
Given the choices:
- A. 50
- B. 289
- C. 210
- D. 1220
The closest choice to the approximate value of 209.97 is C. 210.
Therefore, the approximate value of [tex]\( P \)[/tex] is 210.
