To determine the length [tex]\(L\)[/tex] of a pendulum given that the time [tex]\(T\)[/tex] for one full swing is 2.2 seconds, we start with the formula that relates the two variables:
[tex]\[
T = 2 \pi \sqrt{\frac{L}{32}}
\][/tex]
Given:
- [tex]\(T = 2.2\)[/tex] seconds
- [tex]\(\pi = 3.14\)[/tex]
First, we need to solve this equation for [tex]\(L\)[/tex]. Here are the steps to isolate [tex]\(L\)[/tex].
1. Rearrange the formula to solve for [tex]\(\sqrt{\frac{L}{32}}\)[/tex]:
[tex]\[
\frac{T}{2 \pi} = \sqrt{\frac{L}{32}}
\][/tex]
2. Substitute the values for [tex]\(T\)[/tex] and [tex]\(\pi\)[/tex]:
[tex]\[
\frac{2.2}{2 \times 3.14} = \sqrt{\frac{L}{32}}
\][/tex]
3. Calculate [tex]\(\frac{2.2}{2 \times 3.14}\)[/tex]:
[tex]\[
\frac{2.2}{6.28} \approx 0.350318
\][/tex]
4. Square both sides to remove the square root:
[tex]\[
\left(0.350318\right)^2 = \frac{L}{32}
\][/tex]
[tex]\[
0.122723 \approx \frac{L}{32}
\][/tex]
5. Multiply both sides of the equation by 32 to solve for [tex]\(L\)[/tex]:
[tex]\[
L = 32 \times 0.122723
\][/tex]
[tex]\[
L \approx 3.92714
\][/tex]
The length of the pendulum is approximately 3.927 feet.
Given the possible answers of 2 feet, 4 feet, 11 feet, and 19 feet, the closest value to our calculated length is 4 feet.
Thus, the length of the pendulum that makes one full swing in 2.2 seconds is approximately 4 feet.
