Skate Park Ramp Gradient
A skate park designer models a ramp profile with the curve y = 0.02x³ - 0.3x² + 1.2x for 0 ≤ x ≤ 8, where x and y are in metres. The ramp starts at ground level (0, 0) and must connect smoothly to a flat platform at x = 8. A skater of mass 70 kg will use the ramp, and the designer claims the steepest gradient is safe because it is below 1 (45°). A council engineer is concerned about two things: whether the gradient anywhere exceeds 1, and whether the ramp actually finishes at a sensible height. The designer also wants to know where the steepest point occurs so a warning sign can be placed.
1. Find the maximum gradient of the ramp and the x-value where it occurs. Determine whether the designer's claim that the gradient stays below 1 is correct.
Read the worked answer
Differentiate: y' = 0.06x² - 0.6x + 1.2.
This is a quadratic opening upward, so its minimum occurs at x = 0.6/(2×0.06) = 5. The minimum gradient is y'(5) = 0.06(25) - 0.6(5) + 1.2 = 1.5 - 3 + 1.2 = -0.3.
The maximum gradient on [0, 8] occurs at an endpoint since the parabola opens upward. Check: y'(0) = 1.2, y'(8) = 0.06(64) - 0.6(8) + 1.2 = 3.84 - 4.8 + 1.2 = 0.24.
So the maximum gradient is 1.2 at x = 0, which exceeds 1. The designer's claim is incorrect: the ramp is steepest at the very start, with gradient 1.2 (about 50°).
A warning sign should be placed near x = 0, not at the middle. The negative gradient near x = 5 means the ramp dips downward there, which may itself be a design concern.
2. Find the height of the ramp at x = 8 and assess whether it connects sensibly to a flat platform.
Read the worked answer
Evaluate y(8) = 0.02(512) - 0.3(64) + 1.2(8) = 10.24 - 19.2 + 9.6 = 0.64 m.
The ramp ends only 0.64 m above ground, which is a very low platform height for a skate ramp. Also, the gradient at x = 8 is 0.24, not 0, so the curve does not meet the platform smoothly; there is a kink of about 13.5°.
Judgement: the model does not satisfy the design brief. A smooth join requires y'(8) = 0, which this cubic does not give. The designer should either adjust coefficients or extend the domain to where the gradient is zero, and reconsider the platform height.