Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex], given that [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex], we can follow these steps:
1. Substitute Known Values:
We have the equation [tex]\( f(t) = P e^{rt} \)[/tex]. We are given [tex]\( f(4) = 246.4 \)[/tex], [tex]\( r = 0.04 \)[/tex], and [tex]\( t = 4 \)[/tex]. So, we substitute these values into the equation:
[tex]\[
246.4 = P \times e^{0.04 \times 4}
\][/tex]
2. Calculate the Exponential Part:
Calculate [tex]\( e^{0.04 \times 4} \)[/tex]. This can be done using a scientific calculator or an exponential function in mathematical software:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
Now, solve the equation for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735}
\][/tex]
[tex]\[
P \approx 209.97
\][/tex]
4. Round or Choose the Closest Value:
Looking at the options provided:
A. 50
B. 210
C. 289
D. 1220
The calculated value [tex]\( P \approx 209.97 \)[/tex] is closest to option B, 210.
Therefore, the approximate value of [tex]\( P \)[/tex] is 210.
1. Substitute Known Values:
We have the equation [tex]\( f(t) = P e^{rt} \)[/tex]. We are given [tex]\( f(4) = 246.4 \)[/tex], [tex]\( r = 0.04 \)[/tex], and [tex]\( t = 4 \)[/tex]. So, we substitute these values into the equation:
[tex]\[
246.4 = P \times e^{0.04 \times 4}
\][/tex]
2. Calculate the Exponential Part:
Calculate [tex]\( e^{0.04 \times 4} \)[/tex]. This can be done using a scientific calculator or an exponential function in mathematical software:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
3. Solve for [tex]\( P \)[/tex]:
Now, solve the equation for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735}
\][/tex]
[tex]\[
P \approx 209.97
\][/tex]
4. Round or Choose the Closest Value:
Looking at the options provided:
A. 50
B. 210
C. 289
D. 1220
The calculated value [tex]\( P \approx 209.97 \)[/tex] is closest to option B, 210.
Therefore, the approximate value of [tex]\( P \)[/tex] is 210.
