2026 AHS P1 Q11

2026 AHS P1 Q11

Secondary 4
7 marks

A triangular prism has its base an equilateral triangle of side \(x\text{ cm}\) and a height of \(y\text{ cm}\). The total surface area of the prism is \(30\text{ cm}^2\).

  1. Show that \(y=\dfrac{60-\sqrt3x^2}{6x}\).[2]
  2. Hence show that the volume, \(V\text{ cm}^3\), is given by the equation \(V=\dfrac{5\sqrt3x}{2}-\dfrac{x^3}{8}\).[1]
  3. Given that \(x\) can vary, find the stationary value of \(V\) and determine its nature.[4]

Solution:

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Answer:(a) \(y=\dfrac{60-\sqrt3x^2}{6x}\). (b) \(V=\dfrac{5\sqrt3x}{2}-\dfrac{x^3}{8}\). (c) Maximum volume \(\dfrac{10}{3}\sqrt{5\sqrt3}\text{ cm}^3\approx9.81\text{ cm}^3\).

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