A plastic brick is made in the shape of a right triangular prism. The triangular end is an equilateral triangle with side length cm and the length of the brick is cm.
 

The volume of the brick is 1000 cm³.

1. Find an expression for in terms of .   (2 marks)
   

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2. Show that the total surface area, cm², of the brick is given by
3.       (2 marks)
   

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3. Find the value of for which the brick has minimum total surface area. (You do not have to find this minimum.)   (3 marks)
   

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A cylinder fits exactly in a right circular cone so that the base of the cone and one end of the cylinder are in the same plane as shown in the diagram below. The height of the cone is 5 cm and the radius of the cone is 2 cm.

The radius of the cylinder is cm and the height of the cylinder is cm.

For the cylinder inscribed in the cone as shown above

1. find in terms of .   (2 marks)
   

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The total surface area, cm², of a cylinder of height cm and radius cm is given by the formula

.

2. find in terms of .   (1 mark)
   

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3. find the value of for which is a maximum.   (2 marks)
   

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On 1 January 2010, Tasmania Jones was walking through an ice-covered region of Greenland when he found a large ice cylinder that was made a thousand years ago by the Vikings.

A statue was inside the ice cylinder. The statue was 1 m tall and its base was at the centre of the base of the cylinder.

The cylinder had a height of metres and a diameter of metres. Tasmania Jones found that the volume of the cylinder was 216 m³. At that time, 1 January 2010, the cylinder had not changed in a thousand years. It was exactly as it was when the Vikings made it.

1. Write an expression for in terms of .   (2 marks)
   

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2. Show that the surface area of the cylinder excluding the base, square metres, is given by the rule  .   (1 mark)
   

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Tasmania found that the Vikings made the cylinder so that is a minimum.

3. Find the value of for which is a minimum and find this minimum value of .   (2 marks)
   

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4. Find the value of when is a minimum.   (1 mark)
   

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