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 the solution is

Explanation:

Step1: Identify the triangle parts

The vertical side is 25 ft, and it's split into two equal parts (since supports meet at the center), so each right triangle has a vertical leg of $\frac{25}{2} = 12.5$ ft. The angle with the vertical is $65^\circ$, and $x$ is the hypotenuse.

Step2: Use cosine function

In a right triangle, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 65^\circ$, adjacent = 12.5 ft, hypotenuse = $x$. So $\cos(65^\circ) = \frac{12.5}{x}$.

Step3: Solve for x

Rearrange the formula: $x = \frac{12.5}{\cos(65^\circ)}$. Calculate $\cos(65^\circ) \approx 0.4226$. Then $x \approx \frac{12.5}{0.4226} \approx 29.6$.

Answer:

29.6