Question

a volleyball player hits the ball when it is h feet above the ground. the volleyball net is 8 feet tall. the ball lands 15 feet away from the base of the net, as shown. what is the height of the ball when the player hits it? feet

Explanation:

Step1: Use the Pythagorean theorem

Let's consider the right - triangle formed by the height of the ball when hit (\(h\)), the horizontal distance from the hitting point to the landing point (\(15\) feet), and the hypotenuse (\(20.825\) feet). According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = h\), \(b = 15\), and \(c=20.825\). So we have \(h^{2}+15^{2}=20.825^{2}\).

Step2: Calculate the values

First, calculate \(15^{2}=225\) and \(20.825^{2}=(20 + 0.825)^{2}=20^{2}+2\times20\times0.825+0.825^{2}=400 + 33+0.680625 = 433.680625\).
Then the equation becomes \(h^{2}+225 = 433.680625\).

Step3: Solve for \(h^{2}\)

Subtract \(225\) from both sides of the equation: \(h^{2}=433.680625 - 225\). So \(h^{2}=208.680625\).

Step4: Solve for \(h\)

Take the square root of both sides. Since \(h>0\), \(h=\sqrt{208.680625}=14.45\) feet.

Answer:

\(14.45\)