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? x ≈ feet the solution is

Explanation:

Step1: Find the length of the adjacent side to the angle

Since the footbridge is 20 - foot long and the diagonals meet in the center, the length of the adjacent side to the \(65^{\circ}\) angle (half of the footbridge length) is \(\frac{20}{2}=10\) feet.

Step2: Use the cosine function

We know that \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 65^{\circ}\), adjacent \(= 10\) feet and hypotenuse \(=x\). So, \(\cos(65^{\circ})=\frac{10}{x}\).

Step3: Solve for \(x\)

$$x=\frac{10}{\cos(65^{\circ})}$$

We know that \(\cos(65^{\circ})\approx0.4226\). Then \(x=\frac{10}{0.4226}\approx23.7\)

Answer:

\(23.7\)