Wheelchair Ramp and Concrete Volume
The school needs a wheelchair ramp at the auditorium entrance. The door threshold is 3.5 feet above the sidewalk. The building code says a ramp's slope may be no steeper than 1:12, meaning 1 foot of rise for every 12 feet of horizontal run. The facilities team proposes a ramp with a 36-foot horizontal run and a width of 5 feet. A parent notes that the proposed ramp 'looks long enough' because 36 feet is more than 12 times 3 feet. The team also plans to pour the ramp as a solid triangular prism and needs to estimate concrete volume. For safety, the ramp surface will have railings along both sloped edges. The principal wants to know whether the 36-foot design passes code, what horizontal run would be required if it does not, and how much concrete the compliant ramp needs.
1. Does the proposed 36-foot ramp meet the 1:12 slope code? If not, what horizontal run is required?
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Slope is rise divided by run. The proposed ramp has slope 3.5/36 ≈ 0.0972. The code limit is 1/12 ≈ 0.0833. Since 0.0972 > 0.0833, the 36-foot ramp is too steep and fails code.
The parent's comparison is flawed: 36 feet is 12 × 3 feet, but the rise is 3.5 feet, not 3 feet. For a 3.5-foot rise at the maximum allowed slope, the run must be 3.5 × 12 = 42 feet. The design needs at least 42 feet of horizontal run, so it is 6 feet short.
2. How much concrete is needed for the compliant ramp? Model it as a triangular prism with width 5 feet.
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For the compliant ramp, use a triangular prism with a right-triangle cross section. The triangle has base 42 feet (run) and height 3.5 feet (rise), and the prism width is 5 feet.
Area of triangle = (1/2)(42)(3.5) = 73.5 square feet. Volume = 73.5 × 5 = 367.5 cubic feet.
If the school instead poured the noncompliant 36-foot ramp, the volume would be (1/2)(36)(3.5)(5) = 315 cubic feet, but that design fails the slope requirement. So the compliant ramp needs 367.5 cubic feet of concrete, before accounting for waste or a landing.
3. Find the total length of railing needed for both sloped edges of the compliant ramp. Include a 10% waste allowance.
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The sloping surface is the hypotenuse of a right triangle with legs 42 feet and 3.5 feet. Length = √(42^2 + 3.5^2) = √(1,764 + 12.25) = √1,776.25 ≈ 42.15 feet.
Railings on both sides need about 2 × 42.15 = 84.3 feet of railing. If the team adds 10% for cuts and waste, 84.3 × 1.10 ≈ 92.7 feet, so ordering about 93 feet is reasonable.
The ramp surface is only slightly longer than the horizontal run because the rise is small compared with the run.