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Numeracy Higher November 2016 Unit 2 Q9

WJECHigherCurrent spec8 marksSimilar ShapesVolume & Surface Area

9. A metal round-headed nail can be thought of as a cone sitting on top of a cylinder, which sits on top of a hemisphere.
A company produces round-headed nails of different sizes, but made of the same metal.
Each nail has the following dimensions:

  • height of cone = \(9r\),
  • height of cylinder = \(15r\),
  • radius of the hemisphere = \(12r\),

where \(r\) is the radius of the cylinder and the base radius of the cone.

Nail: cone of height 9r on a cylinder of height 15r on a hemisphere of radius 12r

Diagram not drawn to scale

A metal cuboid of volume 18 000 mm3 is melted down, and re-cast to form round-headed nails of size A, where \(r = 0{\cdot}4\) mm.

From the formula list: Volume of sphere \(= \dfrac{4}{3}\pi r^3\);   Volume of cone \(= \dfrac{1}{3}\pi r^2 h\)

(a) Calculate the greatest number of round-headed nails of size A that can be produced. [6]
(b) Circle either TRUE or FALSE for each statement given below. [2]
STATEMENT
A nail double the height of a size A nail will have a total height of 28·8 mm.TRUEFALSE
A nail double the height of a size A nail will be 8 times the weight of a size A nail.TRUEFALSE
A nail 3 times the height of a size A nail will have a total surface area 6 times that of a size A nail.TRUEFALSE
When \(r = 0{\cdot}8\) mm, the number of nails that could be produced from the same metal cuboid will be double the number of size A nails.TRUEFALSE
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