2025 SST PRELIMS P2 Q9

A packing company's robotic arm has upper arm \(PQ=30\) cm, fixed at \(45.8^\circ\) to the horizontal, and forearm \(QR=17\) cm. The elbow angle is \(\theta\). The tray at \(R\) is 35 cm horizontally and 12 cm vertically above \(X\).

1. Calculate \(\theta\), which allows the arm to pick up the tray.[2]
2. Calibration may change \(\theta\) by \(1^\circ\). Show that this produces a 0.297 cm difference from the intended position of \(R\).[1]
3. The company is comparing the following systems.[7]

   System	Description
   A (Original)	Uses pre-programmed \(\theta\); a 1° error occurs with probability 40% and causes a miss.
   B (Real-time)	Uses camera feedback to recalculate \(\theta\) for every tray; the angle is always accurate.
   C (Hybrid)	Uses pre-programmed \(\theta\); on detecting a 1° error, camera feedback recalculates it.

   The original system handles 300 trays per hour. Each success earns $0.50; each miss costs $0.30. Camera feedback adds 0.4 seconds per affected tray and costs $12 000 per month. The company operates 160 hours per month.
   Determine which system the company should adopt. Justify your decision with clear calculations.