An industrial robot manufacturing company is tasked to design humanoid robots to be used in warehouses where the robots need to pick items from a stack of shelves. The height of the topmost shelf from the ground is 7 feet. To operate, the robot has to move on a track, running parallel to the stack of shelves. The track is fixed 1 foot away from the base of the stack of shelves. Further, the robot cannot bend its arms by more then 60o from the horizontal plane.
If the robot's arms are attached to its shoulder, what should be the minimum height of the robot from the ground to the shoulder for its arms to reach the topmost shelf?
Answer: E
Asked: the minimum shoulder height that lets the arm reach a shelf 7 ft high, 1 ft away horizontally, when the arm can rise at most 60° above the horizontal.
- The shoulder is 1 ft (horizontally) from the shelf face.
- An arm at angle \(\theta\) gains height \(1 \times \tan\theta\) over that 1 ft.
- The steepest allowed angle is 60°, so the greatest gain is \(\tan 60^\circ = \sqrt{3}\) ft.
- Shoulder height \(h\) must satisfy \(h + \sqrt{3} \ge 7\), so \(h \ge 7 - \sqrt{3}\) ft (about 5.27 ft).
Answer: \(7 - \sqrt{3}\) feet