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Answer :
To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P e^{rt} \)[/tex], given that [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex], follow these steps:
1. Substitute the known values into the equation:
We know:
- [tex]\( t = 4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( f(4) = 246.4 \)[/tex]
So, we have the equation:
[tex]\[
f(4) = P \cdot e^{0.04 \cdot 4}
\][/tex]
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]
2. Solve for [tex]\( P \)[/tex]:
To find [tex]\( P \)[/tex], you need to isolate [tex]\( P \)[/tex] on one side of the equation. This can be done by dividing both sides by [tex]\( e^{0.16} \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
3. Calculate [tex]\( e^{0.16} \)[/tex]:
The value of [tex]\( e^{0.16} \)[/tex] is a constant that can be calculated using a calculator. For our purpose, it approximately equals 1.17351.
4. Compute the value of [tex]\( P \)[/tex]:
Now substitute the approximate value of [tex]\( e^{0.16} \)[/tex] into the equation:
[tex]\[
P \approx \frac{246.4}{1.17351} \approx 209.97
\][/tex]
5. Determine the closest value:
Comparing our calculated [tex]\( P \approx 209.97 \)[/tex] to the provided options, the closest approximate value is:
A. 210
So, the approximate value of [tex]\( P \)[/tex] is 210.
1. Substitute the known values into the equation:
We know:
- [tex]\( t = 4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( f(4) = 246.4 \)[/tex]
So, we have the equation:
[tex]\[
f(4) = P \cdot e^{0.04 \cdot 4}
\][/tex]
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]
2. Solve for [tex]\( P \)[/tex]:
To find [tex]\( P \)[/tex], you need to isolate [tex]\( P \)[/tex] on one side of the equation. This can be done by dividing both sides by [tex]\( e^{0.16} \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
3. Calculate [tex]\( e^{0.16} \)[/tex]:
The value of [tex]\( e^{0.16} \)[/tex] is a constant that can be calculated using a calculator. For our purpose, it approximately equals 1.17351.
4. Compute the value of [tex]\( P \)[/tex]:
Now substitute the approximate value of [tex]\( e^{0.16} \)[/tex] into the equation:
[tex]\[
P \approx \frac{246.4}{1.17351} \approx 209.97
\][/tex]
5. Determine the closest value:
Comparing our calculated [tex]\( P \approx 209.97 \)[/tex] to the provided options, the closest approximate value is:
A. 210
So, the approximate value of [tex]\( P \)[/tex] is 210.
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