Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P \cdot e^{rt} \)[/tex] when [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex], follow these steps:
1. Write Down the Given Equation:
You have the equation [tex]\( f(t) = P \cdot e^{rt} \)[/tex].
2. Substitute the Given Values:
Substitute [tex]\( t = 4 \)[/tex], [tex]\( f(4) = 246.4 \)[/tex], and [tex]\( r = 0.04 \)[/tex] into the equation:
[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]
3. Simplify the Exponential Term:
Calculate the exponent: [tex]\( 0.04 \cdot 4 = 0.16 \)[/tex], so the equation becomes:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
Divide both sides of the equation by [tex]\( e^{0.16} \)[/tex] to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
When you compute this division, you find that:
[tex]\[
P \approx 209.97
\][/tex]
5. Select the Closest Option:
The closest option to [tex]\( 209.97 \)[/tex] from the choices provided is [tex]\( 210 \)[/tex].
Therefore, the approximate value of [tex]\( P \)[/tex] is:
B. 210
1. Write Down the Given Equation:
You have the equation [tex]\( f(t) = P \cdot e^{rt} \)[/tex].
2. Substitute the Given Values:
Substitute [tex]\( t = 4 \)[/tex], [tex]\( f(4) = 246.4 \)[/tex], and [tex]\( r = 0.04 \)[/tex] into the equation:
[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]
3. Simplify the Exponential Term:
Calculate the exponent: [tex]\( 0.04 \cdot 4 = 0.16 \)[/tex], so the equation becomes:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
Divide both sides of the equation by [tex]\( e^{0.16} \)[/tex] to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
When you compute this division, you find that:
[tex]\[
P \approx 209.97
\][/tex]
5. Select the Closest Option:
The closest option to [tex]\( 209.97 \)[/tex] from the choices provided is [tex]\( 210 \)[/tex].
Therefore, the approximate value of [tex]\( P \)[/tex] is:
B. 210
