Skate ramp compliance and plywood order
A youth centre is building a skate ramp against the end wall of its hall. The design is a right-angled wedge: the platform is 1.35 m above the floor, the horizontal run from the floor to the foot of the platform is 4.5 m, and the ramp is 2.4 m wide. The platform deck is 1.50 m deep and must be clad in the same plywood as the sloping surface. The two triangular side panels and the vertical face below the platform must also be clad. Plywood comes in sheets measuring 2.40 m by 1.20 m, and the builder budgets 12% extra for cuts and overlaps. Health and safety rules for the hall cap the angle of any ramp at 15° to the horizontal, and the insurer will check before the hall reopens. A volunteer has already cut the two side panels to the 4.5 m run, so any change means recutting them.
1. Check whether the ramp meets the 15° limit and find the minimum horizontal run that would comply with the 1.35 m platform.
Read the worked answer
The side profile is a right triangle with opposite 1.35 m (rise) and adjacent 4.5 m (run).
tan θ = 1.35/4.5 = 0.300 θ = tan⁻¹(0.300) = 16.7°
16.7° > 15°, so the ramp as built fails the rule and the hall cannot reopen with it.
Minimum compliant run: solve tan 15° = 1.35/run run = 1.35/tan 15° = 1.35/0.26795 = 5.04 m
The run must be at least 5.04 m, about 0.54 m longer than the 4.5 m already cut. Alternatively the platform could be lowered, but the design brief fixes it at 1.35 m.
2. Calculate the plywood needed for the ramp as built, including 12% waste, and state how many 2.40 m × 1.20 m sheets to order.
Read the worked answer
Slope length = √(4.5² + 1.35²) = √(20.25 + 1.8225) = √22.0725 = 4.698 m.
Faces to clad:
- Sloping surface: 4.698 × 2.4 = 11.28 m²
- Two triangular sides: 2 × ½ × 4.5 × 1.35 = 6.08 m²
- Vertical face under the platform: 1.35 × 2.4 = 3.24 m²
- Platform deck: 1.50 × 2.4 = 3.60 m²
Total = 11.28 + 6.08 + 3.24 + 3.60 = 24.20 m² With 12% waste: 24.20 × 1.12 = 27.10 m²
One sheet covers 2.40 × 1.20 = 2.88 m². 27.10/2.88 = 9.41 → order 10 sheets.
The judgement point is which faces count: the floor is not clad, but the vertical face and the deck both are.
3. If the run is extended to the compliant minimum, find the change in plywood needed and whether the sheet order changes.
Read the worked answer
New run = 5.038 m (keeping the 1.35 m rise).
New slope length = √(5.038² + 1.35²) = √(25.384 + 1.8225) = √27.206 = 5.216 m. Sloping surface = 5.216 × 2.4 = 12.52 m² (was 11.28 m², +1.24 m²).
New side triangles = 2 × ½ × 5.038 × 1.35 = 6.80 m² (was 6.08 m², +0.72 m²).
Vertical face and deck are unchanged: 3.24 + 3.60 = 6.84 m².
New total = 12.52 + 6.80 + 6.84 = 26.16 m² With 12% waste: 26.16 × 1.12 = 29.30 m² 29.30/2.88 = 10.17 → 11 sheets.
So compliance costs one extra sheet: the long side panels grow as well as the slope, which is the part people forget. The waste allowance is what pushes 10.17 over the sheet boundary.