To find the length \( x \) of the diagonal support, we can set up the problem using trigonometric relationships. Each diagonal support forms a right triangle with the vertical support and the horizontal line of the footbridge.
Given:
- Angle with the vertical, \( \theta = 65^\circ \)
- Half the length of the footbridge (the base of the right triangle), \( \frac{25}{2} = 12.5 \text{ feet} \)
In the right triangle formed, we can use the sine function or the cosine function to relate the angle, the opposite side (vertical support), and the hypotenuse (the diagonal support).
Using the cosine function: \[ \cos(65^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12.5}{x} \]
Rearranging the formula to solve for \( x \): \[ x = \frac{12.5}{\cos(65^\circ)} \]
Now we need to calculate \( \cos(65^\circ) \). Using a calculator, we find: \[ \cos(65^\circ) \approx 0.4226 \]
Now substituting back into the equation for \( x \): \[ x \approx \frac{12.5}{0.4226} \approx 29.6 \text{ feet} \]
Thus, the length of each diagonal support \( x \) is approximately \( \text{29.6 feet} \).
To the nearest tenth of a foot, the answer is: \[ \boxed{29.6 \text{ feet}} \]