You are building a ramp that must cover a horizontal distance of exactly 9 feet. The angle of the ramp from the ground is \( 23^{\circ} \). Determine the length of the ramp, in feet. Round to two decimal places as needed. Show your work here

To find the length of the ramp, we can use a basic trigonometric relationship involving the angle and the horizontal distance. The horizontal distance (the base) is related to the length of the ramp (the hypotenuse) and the angle using the cosine function: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \] Here, the adjacent side is the horizontal distance (9 feet), and θ is the angle of the ramp (23 degrees). Rearranging the equation gives us: \[ \text{hypotenuse} = \frac{\text{adjacent}}{\cos(\theta)} = \frac{9}{\cos(23^{\circ})} \] Now, calculating: \[ \cos(23^{\circ}) \approx 0.9205 \] Thus, \[ \text{hypotenuse} \approx \frac{9}{0.9205} \approx 9.78 \text{ feet} \] So, the length of the ramp is approximately **9.78 feet** when rounded to two decimal places.