Choosing a Loading Ramp Under Two Constraints
Two workers must slide a 300-pound crate up a ramp into a truck bed 3 feet above the ground. Rolling and sliding friction between crate and ramp adds a constant 40 pounds of resistance along the ramp, regardless of the ramp's angle. Three ramps are available: 6, 9, and 12 feet long. Company policy limits any single worker to pushing 75 pounds; two workers may push together and their efforts add. The loading area is only 10 feet deep, so any ramp longer than 10 feet blocks the aisle and is not allowed to be set up.
1. Compute the push required along each ramp, and check which ramps a single worker could use.
Read the worked answer
Without friction, the push along a ramp equals the weight times the height-to-length ratio: F = W × (h/L) = 300 × 3/L = 900/L.
| Ramp | 900/L | + friction 40 lb | Total push |
|---|---|---|---|
| 6 ft | 150 lb | +40 | 190 lb |
| 9 ft | 100 lb | +40 | 140 lb |
| 12 ft | 75 lb | +40 | 115 lb |
No ramp gets within the 75-pound one-worker limit - even the 12-footer needs 115 lb. A single worker cannot do this job with any of the three ramps.
2. Which ramp satisfies the 75-pound-per-worker limit with two workers, and which ones fail?
Read the worked answer
Divide each total by two and compare with 75 lb:
- 6 ft: 190 / 2 = 95 lb each - fails (over 75)
- 9 ft: 140 / 2 = 70 lb each - passes
- 12 ft: 115 / 2 = 57.5 lb each - passes
So the 6-foot ramp is ruled out outright; two workers still exceed the policy limit. The 9-foot and 12-foot ramps both meet the force limit.
3. Which ramp should actually be used? Show that both constraints are being checked, not just the force.
Read the worked answer
The 12-footer meets the force limit but is 12 ft long in a space 10 ft deep - it blocks the aisle, so it is not an option regardless of how easy the push is.
The 9-foot ramp is the only one that clears both constraints at once:
- Force: 140 lb total ÷ 2 = 70 lb per worker ≤ 75 lb ✓
- Fit: 9 ft ≤ 10 ft depth ✓
The 6-footer is shorter and fits but needs 95 lb per worker, which exceeds the limit. Judgement: a ramp problem has more than one failure mode - effort and geometry both have to pass, and the answer is the option that satisfies the tighter of the two.