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Mensuration (Pyramid, Cones and Spheres) - Volume and Surface Area of Pyramids

Mensuration (Pyramid, Cones and Spheres) - Volume and Surface Area of Pyramids

IP 2
6 marks

The diagram shows a candle in the shape of a pyramid \(ABCDE\). \(ABCD\) is a square of side \(6\text{ cm}\) and \(AE = BE = CE = DE = 10\text{ cm}\).

  1. Find the total surface area of the candle.

    [3]

  2. The vertical height of the candle is \(9.06\) cm.

    The candle is melted to form hemispherical candles with radius, \(r\) cm.

    Show that \(r = 3.73\) cm.

    [3]

Editorial note for part (b): the number of hemispherical candles is not specified. The requested radius \(3.73\text{ cm}\) applies to one hemisphere, assuming no wax is lost. For \(n\) equal hemispheres, the radius depends on \(n\), as shown in the solution.

Solution:

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Answer:(a) \(36+12\sqrt{91}\text{ cm}^2\approx1.50\times10^2\text{ cm}^2\) (3 s.f.). (b) \(r\approx3.73\text{ cm}\) for one hemisphere with no wax lost; for \(n\) equal hemispheres, \(r=\sqrt[3]{163.08/(n\pi)}\text{ cm}\).

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