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2024 MI PRU2 P1 Q6

2024 MI PRU2 P1 Q6

7 marks

A closed triangular prism has length \(l\) cm and an equilateral triangle of side \(x\) cm as its base.

  1. Given that the total surface area of this prism is 150 cm\({}^2\), show that the volume \(V\) cm\({}^3\) of the prism is given by
    \[V = \frac{1}{8}x\left(100\sqrt{3} - x^2\right).\][3]
  2. Hence using differentiation, find the value of the maximum volume of the prism, proving that it is a maximum.[4]

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Answer:(i) \(V=\dfrac18x(100\sqrt3-x^2)\) (ii) \(V_{\max}=\dfrac{250}{3^{3/4}}\text{ cm}^3\approx110\text{ cm}^3\), attained when \(x=\dfrac{10}{3^{1/4}}\text{ cm}\)

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