Answer :
To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex], we are given [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex]. Let's solve for [tex]\( P \)[/tex] step-by-step:
1. Understand the function: The given function is [tex]\( f(t) = P e^{rt} \)[/tex]. We know [tex]\( f(4) = 246.4 \)[/tex], so we can write:
[tex]\[
f(4) = P e^{0.04 \times 4} = 246.4
\][/tex]
2. Simplify the exponent: Calculate the exponent term:
[tex]\[
e^{rt} = e^{0.04 \times 4}
\][/tex]
3. Approximate the exponential value: After evaluating [tex]\( e^{0.16} \)[/tex], we find it to be approximately [tex]\( 1.1735 \)[/tex].
4. Solve for [tex]\( P \)[/tex]: Substitute back into the equation and solve for [tex]\( P \)[/tex]:
[tex]\[
246.4 = P \times 1.1735
\][/tex]
[tex]\[
P = \frac{246.4}{1.1735} \approx 209.97
\][/tex]
The approximate value of [tex]\( P \)[/tex] is closest to 210. Therefore, the correct answer is:
A. 210
1. Understand the function: The given function is [tex]\( f(t) = P e^{rt} \)[/tex]. We know [tex]\( f(4) = 246.4 \)[/tex], so we can write:
[tex]\[
f(4) = P e^{0.04 \times 4} = 246.4
\][/tex]
2. Simplify the exponent: Calculate the exponent term:
[tex]\[
e^{rt} = e^{0.04 \times 4}
\][/tex]
3. Approximate the exponential value: After evaluating [tex]\( e^{0.16} \)[/tex], we find it to be approximately [tex]\( 1.1735 \)[/tex].
4. Solve for [tex]\( P \)[/tex]: Substitute back into the equation and solve for [tex]\( P \)[/tex]:
[tex]\[
246.4 = P \times 1.1735
\][/tex]
[tex]\[
P = \frac{246.4}{1.1735} \approx 209.97
\][/tex]
The approximate value of [tex]\( P \)[/tex] is closest to 210. Therefore, the correct answer is:
A. 210
