To find the length \( x \) of the diagonal support, we can use trigonometry. Each diagonal support forms a right triangle with the vertical support and the horizontal distance from the vertical support to the point where the diagonal support meets the horizontal line of the footbridge.
The setup is as follows:
- The footbridge is 20 feet long, so each diagonal support connects at the center of the bridge, dividing it into two segments of 10 feet each.
- The angle between the vertical support and the diagonal support is given as \( 65^\circ \).
We can use the relationship in the right triangle formed by the diagonal support, the vertical support, and the horizontal distance from the base of the vertical support to the center.
Using the sine function, we have:
\[ \sin(65^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{x} \]
where \( h \) is the height of the vertical support, and \( x \) is the length of the diagonal support.
The horizontal distance (base of the triangle) from the vertical support to the center (which is half the length of the footbridge) can be determined with the cosine function:
\[ \cos(65^\circ) = \frac{d}{x} \]
where \( d \) is the distance from the vertical support to the center of the footbridge. Given the bridge is 20 feet long, \( d = 10 \) feet.
- First, we can find \( h \) in terms of \( x \):
From the \( \sin \) equation: \[ h = x \cdot \sin(65^\circ) \]
- Now we can find \( x \) using \( d \):
From the \( \cos \) equation: \[ 10 = x \cdot \cos(65^\circ) \]
Rearranging gives us:
\[ x = \frac{10}{\cos(65^\circ)} \]
Now, plug in the value of \( \cos(65^\circ) \):
Calculating \( \cos(65^\circ) \): \[ \cos(65^\circ) \approx 0.4226 , \text{(using a calculator)} \]
Thus, we substitute:
\[ x = \frac{10}{0.4226} \approx 23.6 \text{ feet} \]
Therefore, the length of the diagonal support, rounded to the nearest tenth of a foot, is:
\[ \boxed{23.6} \text{ feet} \]