Skate Ramp Angle, Run, and Side-Panel Material
A volunteer group is building a wooden skate ramp for a town park. The safety code requires the ramp's angle with the ground to be between 15 degrees and 20 degrees. The ramp must rise 3 feet vertically to a platform. The group has a 10-foot-long sheet of plywood for the sloped surface and a 12-foot horizontal strip of level ground for the run-up. They also have one 4-foot by 8-foot sheet of plywood for two identical triangular side panels, and they must waste 10% for cuts. The platform must be level, and the sloped surface must be cut from the 10-foot sheet without splicing. The group wants to know whether the 10-foot sheet is long enough, what range of angles is actually possible, and whether the side-panel material is sufficient.
1. Find the possible ramp angles and horizontal runs given the 10-foot board and the code.
Read the worked answer
Let theta be the ramp angle. The sloped length L and height 3 ft satisfy sin(theta) = 3/L. With L=10, sin(theta) = 0.3, so theta = arcsin(0.3) = 17.46 degrees. The code requires theta between 15 and 20 degrees. Since 17.46 is above 15, the 10-foot board forces theta to be at least 17.46 degrees; the feasible range is 17.46 <= theta <= 20 degrees. The horizontal run is 3/tan(theta). At theta = 17.46 degrees, run = 3/tan(17.46) = 9.53 ft. At theta = 20 degrees, run = 3/tan(20) = 8.24 ft. Both fit within the 12-foot strip, so the run-up is not the constraint; the board length is.
2. Determine whether the 4-foot by 8-foot side-panel sheet is sufficient, including the 10% waste allowance.
Read the worked answer
The triangular side panel has base equal to the horizontal run and height 3 ft, so area = 0.5 x run x 3. At theta = 17.46 degrees, area = 0.5 x 9.53 x 3 = 14.30 sq ft per panel. Two panels need 28.60 sq ft. Adding 10% waste: 28.60 x 1.10 = 31.46 sq ft. One 4-ft by 8-ft sheet is 32 sq ft, so it is just sufficient, with only 0.54 sq ft spare. At theta = 20 degrees, area per panel = 0.5 x 8.24 x 3 = 12.36 sq ft; two panels plus 10% = 27.19 sq ft, comfortably within 32 sq ft. The material works, but the shallowest allowed angle is very tight and leaves little room for cutting error.
3. What if the group used a 12-foot board for the sloped surface instead? Would that change the feasible angle range?
Read the worked answer
With a 12-ft board, sin(theta) = 3/12 = 0.25, so theta = arcsin(0.25) = 14.48 degrees. That is below the code minimum of 15 degrees, so the code would set theta = 15 degrees as the lower bound. The 12-ft board is then longer than needed, and the run at 15 degrees is 3/tan(15) = 11.20 ft, which still fits within the 12-ft strip. However, the 10-ft board is the limiting material in the actual project, so the group should build at an angle between 17.46 and 20 degrees, preferably nearer 20 degrees to conserve side-panel material.