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 raise its arms by more than 60° 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?
A
Step-by-Step Solution
Key idea: This is a max-reach angle of elevation problem. The robot's arm forms the hypotenuse of a right triangle, with the horizontal distance and the vertical reach forming the other two sides.
Step 1: Identify the given values. The topmost shelf is at a height of 7 feet. The horizontal distance from the track to the shelves is 1 foot. The maximum angle of elevation of the arm is 60°.
Step 2: Let the height of the robot's shoulder be . The vertical distance the arm needs to reach from the shoulder to the topmost shelf is .
Step 3: Use the tangent ratio for the right triangle formed by the arm, the horizontal distance, and the vertical reach.
Step 4: Solve for .
feet.
Answer: feet.