To determine the length of the rope, or how many feet the bucket has been lowered, we need to use the formula that relates the period of a pendulum to its length. The period [tex]\( T \)[/tex] of a pendulum is given by the formula:
[tex]\[ T = 2\pi \sqrt{\frac{L}{g}} \][/tex]
Here,
- [tex]\( T \)[/tex] is the period of the pendulum in seconds.
- [tex]\( L \)[/tex] is the length of the rope (or the distance the bucket has been lowered) in feet.
- [tex]\( g \)[/tex] is the acceleration due to gravity, which is approximately [tex]\( 32.2 \, \text{feet/second}^2 \)[/tex].
We are given that the period [tex]\( T \)[/tex] is 4 seconds. Let’s rearrange the formula to solve for [tex]\( L \)[/tex]:
1. Start with the formula for the period:
[tex]\[ T = 2\pi \sqrt{\frac{L}{g}} \][/tex]
2. Isolate the square root term by dividing both sides by [tex]\( 2\pi \)[/tex]:
[tex]\[ \frac{T}{2\pi} = \sqrt{\frac{L}{g}} \][/tex]
3. Square both sides to eliminate the square root:
[tex]\[ \left( \frac{T}{2\pi} \right)^2 = \frac{L}{g} \][/tex]
4. Multiply both sides by [tex]\( g \)[/tex] to solve for [tex]\( L \)[/tex]:
[tex]\[ L = g \left( \frac{T}{2\pi} \right)^2 \][/tex]
Now plug in the values:
- [tex]\( T = 4 \)[/tex] seconds
- [tex]\( g = 32.2 \, \text{feet/second}^2 \)[/tex]
[tex]\[ L = 32.2 \left( \frac{4}{2\pi} \right)^2 \][/tex]
Performing the calculations step-by-step:
1. Calculate [tex]\( 2\pi \)[/tex]:
[tex]\[ 2\pi \approx 6.2832 \][/tex]
2. Divide the period [tex]\( T \)[/tex] by [tex]\( 2\pi \)[/tex]:
[tex]\[ \frac{4}{6.2832} \approx 0.6366 \][/tex]
3. Square the result:
[tex]\[ (0.6366)^2 \approx 0.4053 \][/tex]
4. Multiply by [tex]\( g \)[/tex]:
[tex]\[ L = 32.2 \times 0.4053 \approx 13.0502 \, \text{feet} \][/tex]
Finally, round the answer to the nearest tenth:
[tex]\[ L \approx 13.1 \, \text{feet} \][/tex]
Therefore, the bucket has been lowered approximately 13.1 feet.
