For the following problem use \( T=2 \pi \sqrt{\frac{L}{g}} \). Let \( \pi \approx 3.14 \) and \( g=32 \) feet per second \( { }^{2} \). A child is swinging on a rope 120 feet long over a river swimming hole. How long does it take (in seconds) to complete one swing back and forth?

Given: - \( T = 2 \pi \sqrt{\frac{L}{g}} \) - \( \pi \approx 3.14 \) - \( g = 32 \) feet per second \( { }^{2} \) - \( L = 120 \) feet To find the time it takes for the child to complete one swing back and forth, we need to calculate the period of the swing using the given formula. Substitute the values into the formula: \[ T = 2 \pi \sqrt{\frac{120}{32}} \] Now, we can calculate the time it takes for the child to complete one swing back and forth. Calculate the value by following steps: - step0: Calculate: \(2\times 3.14\sqrt{\frac{120}{32}}\) - step1: Reduce the fraction: \(2\times 3.14\sqrt{\frac{15}{4}}\) - step2: Simplify the root: \(2\times 3.14\times \frac{\sqrt{15}}{2}\) - step3: Reduce the fraction: \(1\times 3.14\sqrt{15}\) - step4: Multiply the terms: \(3.14\sqrt{15}\) - step5: Simplify: \(12.161168\) The time it takes for the child to complete one swing back and forth is approximately 12.16 seconds.