To find the length \( x \) of the diagonal support, we can use trigonometric ratios given the information stated about the bridge and the angles involved.
In this scenario, we have a right triangle where:
- The vertical support (the shortest leg) is half the length of the footbridge since the diagonal supports meet in the center. Therefore, the length of the vertical support is \( \frac{20}{2} = 10 \) feet.
- The angle between the vertical support and the diagonal support is given as \( 65^\circ \).
We can use the sine function, which relates the opposite side (vertical support) to the hypotenuse (diagonal support):
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
For our triangle:
- \( \theta = 65^\circ \)
- Opposite side = 10 feet (vertical support)
- Hypotenuse = \( x \) (length of the diagonal support)
Substituting into the sine function:
\[ \sin(65^\circ) = \frac{10}{x} \]
Rearranging the equation to solve for \( x \):
\[ x = \frac{10}{\sin(65^\circ)} \]
Now we calculate \( \sin(65^\circ) \):
Using a calculator:
\[ \sin(65^\circ) \approx 0.9063 \]
Now, substituting this back into the equation for \( x \):
\[ x = \frac{10}{0.9063} \approx 11.0 \]
Therefore, the length \( x \) of a diagonal support, to the nearest tenth of a foot, is:
\[ \boxed{11.0} \text{ feet} \]
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