Answer :
To find the approximate value of [tex]\( P \)[/tex], we use the function [tex]\( f(t) = P e^{rt} \)[/tex]. We are given that [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex] and [tex]\( t = 4 \)[/tex].
Let's follow the steps to solve for [tex]\( P \)[/tex]:
1. Start with the equation for the function:
[tex]\[
f(t) = P e^{rt}
\][/tex]
2. Plug in the given values:
[tex]\[
f(4) = P e^{0.04 \times 4}
\][/tex]
[tex]\[
246.4 = P e^{0.16}
\][/tex]
3. Calculate [tex]\( e^{0.16} \)[/tex]. This number represents the mathematical constant [tex]\( e \)[/tex] raised to the power of [tex]\( 0.16 \)[/tex]. The calculated value is approximately [tex]\( 1.1735 \)[/tex].
4. Substitute this back into the equation:
[tex]\[
246.4 = P \times 1.1735
\][/tex]
5. Solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( 1.1735 \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735} \approx 209.97
\][/tex]
The approximate value of [tex]\( P \)[/tex] is about [tex]\( 210 \)[/tex].
Among the options provided in the question, the closest value to 209.97 is [tex]\( 210 \)[/tex].
Therefore, the correct answer is:
A. 210
Let's follow the steps to solve for [tex]\( P \)[/tex]:
1. Start with the equation for the function:
[tex]\[
f(t) = P e^{rt}
\][/tex]
2. Plug in the given values:
[tex]\[
f(4) = P e^{0.04 \times 4}
\][/tex]
[tex]\[
246.4 = P e^{0.16}
\][/tex]
3. Calculate [tex]\( e^{0.16} \)[/tex]. This number represents the mathematical constant [tex]\( e \)[/tex] raised to the power of [tex]\( 0.16 \)[/tex]. The calculated value is approximately [tex]\( 1.1735 \)[/tex].
4. Substitute this back into the equation:
[tex]\[
246.4 = P \times 1.1735
\][/tex]
5. Solve for [tex]\( P \)[/tex] by dividing both sides by [tex]\( 1.1735 \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735} \approx 209.97
\][/tex]
The approximate value of [tex]\( P \)[/tex] is about [tex]\( 210 \)[/tex].
Among the options provided in the question, the closest value to 209.97 is [tex]\( 210 \)[/tex].
Therefore, the correct answer is:
A. 210
