Question

a 20-foot-long footbridge has two diagonal supports that meet in the center of the bridge. each support makes a 65° angle with a short vertical support. what is the length x of a diagonal support, to the nearest tenth of a foot?

Explanation:

Step1: Identify the triangle type

We have a right triangle with one angle \(65^\circ\) and the adjacent side to this angle (the half of the bridge length) is \(\frac{20}{2} = 10\) feet. We need to find the hypotenuse \(x\) (the diagonal support). We can use the cosine function, where \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\).

Step2: Apply the cosine formula

The adjacent side is \(10\) feet, \(\theta = 65^\circ\), and hypotenuse is \(x\). So \(\cos(65^\circ)=\frac{10}{x}\). Rearranging for \(x\), we get \(x=\frac{10}{\cos(65^\circ)}\).

Step3: Calculate the value

First, find \(\cos(65^\circ)\approx0.4226\). Then \(x = \frac{10}{0.4226}\approx23.66\). Rounding to the nearest tenth, we get \(23.7\).

Answer:

\(23.7\)