2024 GESS PRELIMS P2 Q6
2024 GESS PRELIMS P2 Q6
11 marks
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.
- show that \(L\), the perimeter of \(ABCD\), can be expressed in the form \(p + q\sin\theta + r\cos\theta\).[2]
- express \(L\) in the form of \(p + R\cos(\theta - \alpha)\), where \(R > 0\) and \(0^\circ < \alpha < 90^\circ\).[3]
- State the maximum value of \(L\), and the corresponding value of \(\theta\).[2]
- Find the value of \(\theta\) if \(L = 25\) m. Explain why a platform with this value of \(L\) is a poor design.[4]
Solution:
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Answer:(a) \(L=20+12\sin\theta+12\cos\theta\) (shown). (b) \(L=20+12\sqrt2\cos(\theta-45^\circ)\). (c) Maximum \(20+12\sqrt2\text{ m}\) at \(45^\circ\). (d) Formal roots \(117.9^\circ,332.1^\circ\); neither is a feasible stairway angle, so no pictured platform has \(L=25\text{ m}\).
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