Cone, sphere and cylinder in the ratio 1 to 2 to 3
Cone, sphere and cylinder in the ratio 1 to 2 to 3
IP 2
8 marks
A sphere of radius \(r\) fits exactly inside a cylinder, touching its curved surface and both ends. A cone has the same base and the same height as the cylinder.
- Show that the volume of the sphere is \(\dfrac{2}{3}\) of the volume of the cylinder.[2]
- Show that the surface area of the sphere equals the curved surface area of the cylinder.[2]
- Show that the volumes of the cone, the sphere and the cylinder are in the ratio \(1:2:3\) for every value of \(r\).[2]
- Three identical balls fit exactly in a cylindrical tin, stacked one above another and touching the curved surface. Use (a) to find the fraction of the tin's volume that is empty, without calculating any volume.[2]
Solution:
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Answer:(a) \(\dfrac{4}{3}\pi r^3:2\pi r^3=2:3\) (b) Both \(4\pi r^2\) (c) \(\dfrac{2}{3}\pi r^3:\dfrac{4}{3}\pi r^3:2\pi r^3=1:2:3\) (d) \(\dfrac{1}{3}\)
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