Question

heather is building a rectangular wooden frame to support a wall. the frame is 8 feet by 3 feet. heather is adding a diagonal beam to the frame. approximately how long should the beam be? between 8 and 9 feet 10 feet between 9 and 10 feet between 11 and 12 feet

Explanation:

Step1: Recall the Pythagorean theorem

For a right - triangle (which is formed by the length, width, and diagonal of a rectangle), the Pythagorean theorem states that \(c^{2}=a^{2}+b^{2}\), where \(a\) and \(b\) are the legs of the right - triangle and \(c\) is the hypotenuse (the diagonal in this case). Here, \(a = 8\) feet and \(b=3\) feet.

Step2: Calculate \(a^{2}+b^{2}\)

First, calculate \(a^{2}\) and \(b^{2}\). \(a^{2}=8^{2}=64\) and \(b^{2}=3^{2} = 9\). Then \(a^{2}+b^{2}=64 + 9=73\).

Step3: Find the square root of 73

We know that \(\sqrt{64}=8\) and \(\sqrt{81}=9\). Since \(64<73<81\), we have \(\sqrt{64}<\sqrt{73}<\sqrt{81}\), which means \(8 < \sqrt{73}<9\). So the length of the diagonal (the beam) is between 8 and 9 feet.

Answer:

Between 8 and 9 feet