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Robot Shoulder Height Calculation

XAT 2025 · Quantitative Ability & Data Interpretation · Medium · Trigonometry

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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?

  1. A.

    3\sqrt{3} feet

  2. B.

    7 feet

  3. C.

    7−37 - \sqrt{3} feet

  4. D.

    None of the other options is correct

  5. E.

    6+36 + \sqrt{3} feet

Answer

C

Explanation

Let hh be the height of the shoulder from the ground. Horizontal distance from shoulder to shelf base = 1 foot. The topmost shelf is at height 7 feet, so the vertical distance from shoulder to topmost shelf is 7−h7 - h. The angle of elevation of the arm from the horizontal plane can be at most 60∘60^\circ. In the right-angled triangle formed: tan⁡(60∘)=Vertical distanceHorizontal distance=7−h1  ⟹  3=7−h  ⟹  h=7−3\tan(60^\circ) = \frac{\text{Vertical distance}}{\text{Horizontal distance}} = \frac{7 - h}{1} \implies \sqrt{3} = 7 - h \implies h = 7 - \sqrt{3}. Thus, minimum shoulder height is 7−37 - \sqrt{3} feet.

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