Answer :
To solve for the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex] when given [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex], we can follow these steps:
1. Understand the given equation: We have the function [tex]\( f(t) = P e^{rt} \)[/tex] where [tex]\( f(4) = 246.4 \)[/tex], [tex]\( r = 0.04 \)[/tex], and [tex]\( t = 4 \)[/tex].
2. Substitute the known values: First, we substitute the given values into the equation:
[tex]\[
246.4 = P e^{0.04 \times 4}
\][/tex]
3. Calculate the exponent: Calculate [tex]\( 0.04 \times 4 \)[/tex] which gives us [tex]\( 0.16 \)[/tex].
4. Find the value of [tex]\( e^{0.16} \)[/tex]: Using a calculator or exponential function tool, compute [tex]\( e^{0.16} \)[/tex]. The value is approximately [tex]\( 1.1735 \)[/tex].
5. Solve for [tex]\( P \)[/tex]: Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735}
\][/tex]
6. Calculate [tex]\( P \)[/tex]: Performing the division gives approximately [tex]\( P \approx 210 \)[/tex].
Therefore, the approximate value of [tex]\( P \)[/tex] is closest to 210.
So, the answer is:
B. 210
1. Understand the given equation: We have the function [tex]\( f(t) = P e^{rt} \)[/tex] where [tex]\( f(4) = 246.4 \)[/tex], [tex]\( r = 0.04 \)[/tex], and [tex]\( t = 4 \)[/tex].
2. Substitute the known values: First, we substitute the given values into the equation:
[tex]\[
246.4 = P e^{0.04 \times 4}
\][/tex]
3. Calculate the exponent: Calculate [tex]\( 0.04 \times 4 \)[/tex] which gives us [tex]\( 0.16 \)[/tex].
4. Find the value of [tex]\( e^{0.16} \)[/tex]: Using a calculator or exponential function tool, compute [tex]\( e^{0.16} \)[/tex]. The value is approximately [tex]\( 1.1735 \)[/tex].
5. Solve for [tex]\( P \)[/tex]: Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{1.1735}
\][/tex]
6. Calculate [tex]\( P \)[/tex]: Performing the division gives approximately [tex]\( P \approx 210 \)[/tex].
Therefore, the approximate value of [tex]\( P \)[/tex] is closest to 210.
So, the answer is:
B. 210
