2026 ZHSS P1 Q8

A delivery company uses three drones, \(P\), \(Q\) and \(R\), for a coordinated flight test.

Drone \(Q\) is hovering at a fixed point 7 m due south of a control point \(O\). Drone \(P\) flies due west from \(O\).

At time \(t\) seconds, drone \(P\) is \(2x\) metres west of \(O\), \(x\) is increasing at a constant rate of 5 m/s.

Drone \(R\) is programmed to continuously maintain an equilateral triangle formation with drones \(P\) and \(Q\) throughout the flight.

1. Show that the distance between the two drones, \(PQ\), is \(\sqrt{4x^2+49}\) metres.[1]
2. Find the rate of change of the distance \(PQ\), in m/s, when \(x=12\).[4]
3. The communication software requires the area enclosed by the three drones to increase at a rate of at least \(250\text{ m}^2/\text{s}\).
   Given that area of \(\triangle PQR=\dfrac12\times PR\times QR\times\sin\angle PRQ\), determine the shortest time taken for drone \(P\) to meet the requirement.[5]