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.

Explanation:

Step1: Recall Pythagorean Theorem

The Pythagorean Theorem states that for a right triangle with legs \(a\), \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). In this problem, we have a right triangle where one leg is \(x\) (the distance we need to find), another leg is \(9.8\) in, and the hypotenuse is \(13.4\) in. So we can rearrange the formula to solve for \(x\): \(x = \sqrt{c^{2}-b^{2}}\) (assuming \(c = 13.4\) and \(b=9.8\)).

Step2: Substitute the values

Substitute \(c = 13.4\) and \(b = 9.8\) into the formula:
\[
x=\sqrt{(13.4)^{2}-(9.8)^{2}}
\]
First, calculate \((13.4)^{2}=13.4\times13.4 = 179.56\) and \((9.8)^{2}=9.8\times9.8=96.04\).

Step3: Calculate the difference

Subtract the two results: \(179.56 - 96.04=83.52\).

Step4: Take the square root

Take the square root of \(83.52\):
\[
x=\sqrt{83.52}\approx9.14
\]

Answer:

The distance between the rods where they meet the backboard is approximately \(9.14\) inches.