Question

1. modeling with mathematics the backboard of the basketball hoop forms a right triangle with the supporting rods, as shown. use the pythagorean theorem (theorem 9.1) to approximate the distance between the rods where they meet the backboard. image of basketball hoop backboard with right triangle, one leg 9.8 in, hypotenuse 13.4 in, and unknown leg x

Explanation:

Step1: Recall Pythagorean Theorem

The Pythagorean Theorem states that for a right triangle with legs \(a\), \(b\) and hypotenuse \(c\), \(a^2 + b^2 = c^2\). Here, the hypotenuse \(c = 13.4\) in and one leg \(a = 9.8\) in, and we need to find the other leg \(x\). So we can rewrite the formula as \(x^2 = c^2 - a^2\).
\[x^2 = 13.4^2 - 9.8^2\]

Step2: Calculate the squares

First, calculate \(13.4^2\) and \(9.8^2\).
\(13.4^2 = 13.4\times13.4 = 179.56\)
\(9.8^2 = 9.8\times9.8 = 96.04\)
Then substitute these values into the equation:
\[x^2 = 179.56 - 96.04 = 83.52\]

Step3: Solve for \(x\)

Take the square root of both sides to find \(x\):
\[x = \sqrt{83.52} \approx 9.14\] (rounded to two decimal places)

Answer:

Approximately \(9.14\) inches