Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex], we are given that [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex].
Step-by-step, we'll solve for [tex]\( P \)[/tex] as follows:
1. Identify the given data:
- For [tex]\( t = 4 \)[/tex], [tex]\( f(4) = 246.4 \)[/tex].
- The rate [tex]\( r = 0.04 \)[/tex].
2. Write the function equation:
[tex]\[
f(t) = P e^{rt}
\][/tex]
3. Substitute the known values into the equation:
[tex]\[
246.4 = P e^{(0.04 \times 4)}
\][/tex]
4. Calculate the exponent:
[tex]\[
e^{(0.04 \times 4)} = e^{0.16}
\][/tex]
5. Estimate [tex]\( e^{0.16} \)[/tex]:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
6. Solve for [tex]\( P \)[/tex]:
[tex]\[
246.4 = P \times 1.1735
\][/tex]
7. Divide both sides by [tex]\( 1.1735 \)[/tex] to find [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]
Based on this calculation, the approximate value of [tex]\( P \)[/tex] is closest to option D, 210.
Step-by-step, we'll solve for [tex]\( P \)[/tex] as follows:
1. Identify the given data:
- For [tex]\( t = 4 \)[/tex], [tex]\( f(4) = 246.4 \)[/tex].
- The rate [tex]\( r = 0.04 \)[/tex].
2. Write the function equation:
[tex]\[
f(t) = P e^{rt}
\][/tex]
3. Substitute the known values into the equation:
[tex]\[
246.4 = P e^{(0.04 \times 4)}
\][/tex]
4. Calculate the exponent:
[tex]\[
e^{(0.04 \times 4)} = e^{0.16}
\][/tex]
5. Estimate [tex]\( e^{0.16} \)[/tex]:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
6. Solve for [tex]\( P \)[/tex]:
[tex]\[
246.4 = P \times 1.1735
\][/tex]
7. Divide both sides by [tex]\( 1.1735 \)[/tex] to find [tex]\( P \)[/tex]:
[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]
Based on this calculation, the approximate value of [tex]\( P \)[/tex] is closest to option D, 210.
