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: Determine the adjacent - side length to the angle

The two diagonal supports meet in the center of the 20 - foot - long bridge. So the length of the adjacent side to the 65° angle for each right - triangle formed is $\frac{20}{2}=10$ feet.

Step2: Use the cosine function

We know that $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 65^{\circ}$ and the adjacent side $a = 10$ feet, and the hypotenuse is $x$. So $\cos65^{\circ}=\frac{10}{x}$.

Step3: Solve for $x$

We can rewrite the equation as $x=\frac{10}{\cos65^{\circ}}$. Since $\cos65^{\circ}\approx0.4226$, then $x=\frac{10}{0.4226}\approx23.66$. Rounding to the nearest tenth, $x\approx23.7$.

Answer:

$23.7$