Answer :
To solve this problem, we need to find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^t \)[/tex], given that [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex].
1. First, note that the variable [tex]\( t \)[/tex] in the function is affected by the rate [tex]\( r \)[/tex]. Here, we are given that [tex]\( t = 4 \times 0.04 \)[/tex].
2. Calculate [tex]\( t \)[/tex] by multiplying:
[tex]\[
t = 4 \times 0.04 = 0.16
\][/tex]
3. Substitute [tex]\( t \)[/tex] into the function:
[tex]\[
f(t) = P e^t = P e^{0.16}
\][/tex]
4. We are given that [tex]\( f(4) = 246.4 \)[/tex]. So, substitute this value into the equation:
[tex]\[
246.4 = P e^{0.16}
\][/tex]
5. To find [tex]\( P \)[/tex], we need to solve the equation for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
6. Calculate [tex]\( e^{0.16} \)[/tex] using the approximate value for [tex]\( e\)[/tex], and then divide 246.4 by this result to find [tex]\( P \)[/tex].
When calculated, this gives an approximate value of:
[tex]\[
P \approx 209.97
\][/tex]
Therefore, the closest answer choice to this value is [tex]\( \text{B. } 210 \)[/tex].
1. First, note that the variable [tex]\( t \)[/tex] in the function is affected by the rate [tex]\( r \)[/tex]. Here, we are given that [tex]\( t = 4 \times 0.04 \)[/tex].
2. Calculate [tex]\( t \)[/tex] by multiplying:
[tex]\[
t = 4 \times 0.04 = 0.16
\][/tex]
3. Substitute [tex]\( t \)[/tex] into the function:
[tex]\[
f(t) = P e^t = P e^{0.16}
\][/tex]
4. We are given that [tex]\( f(4) = 246.4 \)[/tex]. So, substitute this value into the equation:
[tex]\[
246.4 = P e^{0.16}
\][/tex]
5. To find [tex]\( P \)[/tex], we need to solve the equation for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
6. Calculate [tex]\( e^{0.16} \)[/tex] using the approximate value for [tex]\( e\)[/tex], and then divide 246.4 by this result to find [tex]\( P \)[/tex].
When calculated, this gives an approximate value of:
[tex]\[
P \approx 209.97
\][/tex]
Therefore, the closest answer choice to this value is [tex]\( \text{B. } 210 \)[/tex].
