2024 GESS PRELIMS P2 Q6

A competitive platform diving competition involves one or two divers performing spins, twists and somersaults into a pool from a series of heights.

The diagram shows the cross-section of a design of the diving platform, which consists of a trapezium-shaped tower \(ABCD\). \(AB\) is the stairway which the divers use to reach the platform.

Given that \(\angle BAD = \theta\), where \(\theta\) can vary and \(\angle ADC = 90^\circ\), the length of \(AB\) and \(BC\) are \(12\) m and \(4\) m respectively.

1. show that \(L\), the perimeter of \(ABCD\), can be expressed in the form \(p + q\sin\theta + r\cos\theta\).[2]
2. express \(L\) in the form of \(p + R\cos(\theta - \alpha)\), where \(R > 0\) and \(0^\circ < \alpha < 90^\circ\).[3]
3. State the maximum value of \(L\), and the corresponding value of \(\theta\).[2]
4. Find the value of \(\theta\) if \(L = 25\) m. Explain why a platform with this value of \(L\) is a poor design.[4]