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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.

A diagram shows the footbridge, the diagonal supports and the vertical supports create two right triangles. At the top of the diagram a horizontal line is shown and labeled 25 feet. The two right triangles are shown below this horizontal line.The two longest legs of both right triangles touch at the center of the footbridge.The shortest leg of both right triangles represents the vertical support on each end of the footbridge. The hypotenuse of both right triangles form the diagonal supports that meet at the center of the footbridge. The full length of the footbridge is 25 feet. The vertical supports are not labeled. Each diagonal support is labeled X. A 65-degree-angle is formed between the short vertical support and the diagonal support on each end of the bridge.

What is the length x of a diagonal support, to the nearest tenth of a foot?

1 answer

To find the length \( x \) of each diagonal support, we can model the situation using trigonometry. Since the footbridge is 25 feet long and the diagonal supports meet in the center, each right triangle has a base (half the length of the bridge) of 12.5 feet.

We can break this down as follows:

  1. Identify the right triangle:

    • One leg (the horizontal leg) of the right triangle is \( 12.5 \) feet (half of the 25-foot bridge).
    • The angle between the horizontal leg and the hypotenuse (diagonal support) is \( 65^\circ \).
    • The other leg (the vertical leg) of the triangle is the vertical support.

  2. Use trigonometric functions: We can use the cosine of the angle to find the length of the hypotenuse \( x \):

    \[ \cos(65^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12.5}{x} \]

    Rearranging the formula gives:

    \[ x = \frac{12.5}{\cos(65^\circ)} \]

  3. Calculate \( \cos(65^\circ) \): Using a calculator, \[ \cos(65^\circ) \approx 0.4226 \]

  4. Substitute back into the equation: \[ x = \frac{12.5}{0.4226} \approx 29.6 \text{ feet} \]

Thus, the length of each diagonal support \( x \) is approximately 29.6 feet, rounded to the nearest tenth.