A cylinder of height and radius is formed from a thin rectangular sheet of metal of length and , by cutting along the dashed lines shown below.

The volume of the cylinder, in terms of and , is given by

A cone is inscribed in a sphere of radius , centred at . The height of the cone is and the radius of the base is , as shown in the diagram.

1. Show that the volume, , of the cone is given by
2.    .   (2 marks)
   

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2. Find the value of for which the volume of the cone is a maximum. You must give reasons why your value of gives the maximum volume.   (3 marks)
   

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A cylinder of radius    and height    is to be inscribed in a sphere of radius    centred at    as shown.

1. Show that the volume of the cylinder is given by
2.       (1 mark)
   

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2. Hence, or otherwise, show that the cylinder has a maximum volume when
3.       (3 marks)
   

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