2025 AISS PRELIMS P1 Q8

2025 AISS PRELIMS P1 Q8

Secondary 4
10 marks
2025 Ahmad Ibrahim Secondary School Prelims A Math Paper 1

A prototype consists of a cylindrical container of height \(h\) cm and radius \(r\) cm inscribed in a hollow sphere with centre \(O\).

The sphere has a surface area of \(6400\pi\) cm\(^2\) and both the sphere and container have negligible thickness.

  1. Show that the volume of the cylinder container, \(V\) cm\(^3\), is given by[3]
    \[V=2\pi r^2\sqrt{1600-r^2}.\]
  2. Given that \(r\) can vary, find the value of \(r\) for which the volume \(V\) is stationary.[5]
  3. A scientist plans to launch this prototype into outer space carrying as much fuel as possible. Explain whether the prototype can satisfy the scientist’s requirement.[2]

Solution:

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Answer:(a) \(V=2\pi r^2\sqrt{1600-r^2}\) (b) \(r=40\sqrt{\frac{2}{3}}=32.7\text{ cm}\) (c) Yes, this gives the maximum volume.

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