📐 Part A: Mathematical Analysis

1. A quarter-pipe ramp has a radius of 3.5 metres. Calculate the angle of incline at the point where a skater is 2 metres horizontally from the centre.

2. For safety standards, ramps must not exceed a 45° angle. If a launch ramp is 8 metres long horizontally and 6 metres high, does it meet safety requirements?

Yes, angle is approximately 37°

No, angle is approximately 53°

Yes, angle is exactly 45°

Cannot determine from given information

3. Calculate the actual length of the ramp surface (hypotenuse) for the ramp in question 2.

4. A bowl section has a circular cross-section with diameter 12 metres. What is the maximum vertical drop a skater experiences?

Maximum drop: _______ metres

5. Compare the costs: Concrete costs $180 per cubic metre, timber costs $320 per cubic metre. For a ramp requiring 15 cubic metres of material, what is the cost difference?

🏗️ Part B: Design and Safety Analysis

6. Create a scale drawing (1:25 scale) of a launch ramp that is 10m long, 4m wide, and 3m high at the launch point. Show the front elevation view.

7. List three mathematical techniques used in skate park design and explain why each is important for safety.

8. A skater launches from a 4-metre high ramp at 30° angle. Using projectile motion principles, estimate the horizontal distance travelled if initial velocity is 8 m/s.

9. Analyse the practicality: Which ramp design would be most cost-effective while maintaining safety?

Steep concrete ramp (50° angle)

Moderate timber ramp (35° angle)

Gradual concrete ramp (25° angle)

Curved composite ramp (variable angle)

10. Justify your choice from question 9, considering safety standards, material costs, and user experience.

📊 Part C: Reflection and Recommendations

11. Reflect on the mathematical processes used in this analysis. Which technique was most crucial for ensuring skater safety?

12. Based on your calculations, write a brief recommendation report for the council about the proposed skate park design. Include at least two mathematical justifications.