Question

a 25 - 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

Explanation:

Step1: Find half - length of bridge

The bridge is 25 feet long. The two diagonal supports meet in the center, so the base of the right - triangle formed by a diagonal support, the vertical support, and half of the bridge is $\frac{25}{2}=12.5$ feet.

Step2: Use cosine function

We know that $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 65^{\circ}$ and the adjacent side to the $65^{\circ}$ angle is 12.5 feet, and the hypotenuse is $x$. So, $\cos65^{\circ}=\frac{12.5}{x}$.

Step3: Solve for $x$

We can rewrite the equation as $x=\frac{12.5}{\cos65^{\circ}}$. Since $\cos65^{\circ}\approx0.4226$, then $x=\frac{12.5}{0.4226}\approx29.6$.

Answer:

$29.6$