Modelling a skate ramp with a piecewise function
A skate ramp's cross-section is modelled by a function h(d), the height in metres at horizontal distance d metres from the start. For 0 ≤ d ≤ 4 the surface rises as h(d) = 0.25d². For 4 < d ≤ 10 it is a straight descent h(d) = 6 − 0.6(d − 4). A skater notes that at d = 4 the two pieces give different heights, and a second skater says the ramp "must be broken" because the slope changes abruptly. The designer insists the model is fine because each piece is valid on its own interval. A safety rule requires the surface to be continuous and to have no vertical step greater than 0.05 m.
1. Evaluate h at d = 4 from each piece and determine whether the model is continuous there.
Read the worked answer
Left piece: h(4) = 0.25 × 16 = 4.0 m. Right piece: h(4) = 6 − 0.6(0) = 6.0 m.
The two pieces disagree by 2.0 m at the join. Since the rule requires a step no greater than 0.05 m, the model fails badly - there is a 2 m vertical jump. The designer's claim that "each piece is valid on its own interval" is true but irrelevant: continuity is a property of the join, not of the pieces separately.
2. Adjust only the constant term of the second piece so the ramp is continuous at d = 4, then state the new height at d = 10.
Read the worked answer
Keep the gradient −0.6 but shift the intercept. We need h(4) = 4 from the right piece: 4 = c − 0.6(0), so c = 4. New rule: h(d) = 4 − 0.6(d − 4).
Check at d = 10: h(10) = 4 − 0.6(6) = 4 − 3.6 = 0.4 m.
So the ramp now runs continuously from 4.0 m at d = 4 down to 0.4 m at d = 10. Note this changes the end height from the original 6 − 3.6 = 2.4 m to 0.4 m, so the designer must confirm 0.4 m is acceptable at the far end.
3. The second skater objects to the abrupt slope change at d = 4. Quantify the slope on each side and assess whether the objection concerns continuity or smoothness.
Read the worked answer
Left slope at d = 4: derivative of 0.25d² is 0.5d, so at d = 4 the slope is 2.0 (rising steeply). Right slope: constant −0.6 (descending).
The slopes differ by 2.6, a sharp corner. This is a smoothness (differentiability) issue, not a continuity issue - after the fix the surface is unbroken but kinked. The safety rule as stated only demands continuity, so the ramp passes the stated rule. The skater's concern is real for ride quality but is a different criterion from the one written; conflating the two is the trap.