Question

the time, in seconds, that it takes a pendulum to swing back and forth is modeled by the equation below. ( f(l)=2pisqrt{\frac{l}{32}} ), where ( l ) is the length of the pendulum in feet. what is the approximate length of a pendulum that takes ( 2.4pi ) seconds to swing back and forth? 1.72 ft 3.05 ft 38.40 ft 46.08 ft

Explanation:

1. First, set up the equation:

• We are given the formula \(T = 2\pi\sqrt{\frac{l}{32}}\), where \(T\) is the time - period of the pendulum's swing and \(l\) is the length of the pendulum.
• We know that \(T = 2.4\pi\) seconds. So, we substitute \(T\) into the formula:
• \(2.4\pi=2\pi\sqrt{\frac{l}{32}}\).

1. Then, solve for \(l\):

• First, divide both sides of the equation by \(2\pi\):
• \(\frac{2.4\pi}{2\pi}=\sqrt{\frac{l}{32}}\).
• Simplifying the left - hand side gives \(1.2=\sqrt{\frac{l}{32}}\).
• Next, square both sides of the equation to get rid of the square - root:
• \((1.2)^2=\frac{l}{32}\).
• So, \(1.44=\frac{l}{32}\).
• Then, multiply both sides by 32 to solve for \(l\):
• \(l = 1.44\times32=46.08\) feet.

Answer:

46.08 ft