Question

1. a rectangular wooden gate with two diagonal boards across the front is shown below.

image of a rectangle with length 11 feet, width x, and two diagonals
if the length of one of the diagonals is 14 feet, find the value of x, the width of the gate. round to the nearest tenth.

Explanation:

Step1: Identify the shape and formula

The gate is a rectangle, so the diagonal, length, and width form a right triangle. We use the Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c\) is the diagonal, \(b\) is the length, and \(a\) (which is \(x\)) is the width.

Step2: Substitute known values

We know \(c = 14\) feet and \(b = 11\) feet. Substitute into the formula: \(x^2 + 11^2 = 14^2\).

Step3: Solve for \(x^2\)

Calculate \(11^2 = 121\) and \(14^2 = 196\). Then \(x^2 = 196 - 121 = 75\).

Step4: Solve for \(x\)

Take the square root of 75: \(x=\sqrt{75}\approx8.7\) (rounded to the nearest tenth).

Answer:

\(8.7\)