2026 CCHY P1 Q12

2026 CCHY P1 Q12

Secondary 4
9 marks

The diagram below shows a park with perimeter \(ABCD\) where angle \(ABC=\) angle \(ADC=90^\circ\) and angle \(BAD=\theta\) where \(0^\circ<\theta<90^\circ\). \(AB=7\) km and \(BC=5\) km.

  1. Show that the perimeter of the park, \(P\) km, is given by \(P=12+12\sin\theta+2\cos\theta\).[2]
  2. Express \(P\) in the form \(R\cos(\theta-\alpha)+12\), where \(R>0\) and \(0^\circ<\alpha<90^\circ\).[3]

Hence find

  1. the maximum possible perimeter of the park and the corresponding value of \(\theta\),[2]
  2. the value of \(\theta\) such that \(P=23.6\).[2]

Solution:

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Answer:(a) \(P=12+12\sin\theta+2\cos\theta\). (b) \(R=2\sqrt{37},\alpha=80.5^\circ\). (c) 24.2 km at \(80.5^\circ\). (d) \(63.0^\circ\).

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