To determine the length of the swing using the given swing period and the pendulum formula, follow these steps:
1. Identify the given values:
- Period of the swing ([tex]\( T \)[/tex]) = 3.1 seconds
- Acceleration due to gravity ([tex]\( g \)[/tex]) = 32 feet per second squared
2. Recall the pendulum formula:
[tex]\[
T = 2 \pi \sqrt{\frac{L}{g}}
\][/tex]
3. Rearrange the formula to solve for [tex]\( L \)[/tex]:
First, isolate the square root by dividing both sides by [tex]\(2\pi\)[/tex]:
[tex]\[
\frac{T}{2\pi} = \sqrt{\frac{L}{g}}
\][/tex]
Next, square both sides to get rid of the square root:
[tex]\[
\left(\frac{T}{2\pi}\right)^2 = \frac{L}{g}
\][/tex]
Finally, solve for [tex]\( L \)[/tex] by multiplying both sides by [tex]\( g \)[/tex]:
[tex]\[
L = \left(\frac{T}{2\pi}\right)^2 \times g
\][/tex]
4. Substitute the known values into the equation:
[tex]\[
L = \left(\frac{3.1}{2\pi}\right)^2 \times 32
\][/tex]
5. Calculate the value step-by-step:
- Calculate [tex]\(2\pi\)[/tex]:
[tex]\[
2\pi \approx 6.2832
\][/tex]
- Divide [tex]\(T\)[/tex] by [tex]\(2\pi\)[/tex]:
[tex]\[
\frac{3.1}{6.2832} \approx 0.4934
\][/tex]
- Square the result:
[tex]\[
(0.4934)^2 \approx 0.2435
\][/tex]
- Multiply by [tex]\( g \)[/tex] (32):
[tex]\[
0.2435 \times 32 \approx 7.792
\][/tex]
6. Round the final result to the nearest tenth:
[tex]\[
7.792 \approx 7.8
\][/tex]
Therefore, the length of the swing is 7.8 feet.
Thus, the correct answer is:
A. 7.8 feet
