Question

the formula ( t = 2pisqrt{\frac{l}{32}} ) gives the time it takes in seconds, t, for a pendulum to make one full swing back and forth, where l is the length of the pendulum, in feet. to the nearest foot, what is the length of a pendulum that makes one full swing in 1.9 s? use 3.14 for ( pi ).

Explanation:

Step1: Substitute \(T = 1.9\) into the formula

Given \(T = 2\pi\sqrt{\frac{L}{32}}\), substitute \(T=1.9\) and \(\pi = 3.14\). So, \(1.9=2\times3.14\sqrt{\frac{L}{32}}\).

Step2: Simplify the equation

First, calculate \(2\times3.14 = 6.28\). The equation becomes \(1.9 = 6.28\sqrt{\frac{L}{32}}\). Then, divide both sides by \(6.28\): \(\sqrt{\frac{L}{32}}=\frac{1.9}{6.28}\approx0.3025\).

Step3: Square both sides

Squaring both sides gives \(\frac{L}{32}=(0.3025)^2 = 0.0915\).

Step4: Solve for \(L\)

Multiply both sides by \(32\): \(L=0.0915\times32 = 2.928\approx3\).

Answer:

\(3\)