1.  A shop sign is to be created with semi-circular ends as shown in the picture. The shop manager wants the sign to have a border which is made from flexible plastic.

a) What length of border will be required to go round the sign?

b) If the border costs £9.99 per metre, how much will the border cost?

c) The cost of the sign is £150. The shop manager will not be charged just for the cost of the sign and the border, VAT will be added to the total at 17.5%. How much will the shop manager have to pay for the completed sign in total?

a) The length of border going round the sign is the same as the perimeter of the sign (i.e. the distance around it). The distance is 2 x 250cm (for the two horizontal lines) plus the distance around 2 semi-circles.

Distance around 2 semi-circles = distance around a circle (in other words the circumference which is 2pr)

Hence we get length of border = (2 x 250) + 2p x 80

= 10.27m

b) Cost of border = 10.27 x £9.99 = £100.17

c) The bill will look like this:

Cost of border    £100.17
Cost of Sign        £150.00
Sub-Total            £250.17

VAT                       43.78

Total                   £293.95 

Where VAT amount is calculated from £250.17 x 0.175

a) Volume of a cone = ¹/₃pr²  x  height

= ¹/₃p(6)²  x  12

= 144p

= 452.39 cm³

b) Volume of ball (sphere) = ⁴/₃pr³ 

= 85p

= 268.08 cm³

c) To calculate the mass of glass we need to know the volume of the cone - volume of ball.

= 144p - 85p = 59p

Mass = Density x Volume

Mass of glass in cone = 2.85 x 59p = 528.25 g

Mass of Ball = 8.91 x 85p = 2,379.29g

Hence Total Mass = 528.25 + 2,379.29

= 2,907.54 g

3.  The underneath of an arch in a town's railway viaduct is to be sold for a company to use as storage space. Each end of the arch must be boarded up before the arch is sold to allow any damp areas to dry out. The top of the arch forms a semi-circle.

a) What is the total area of boarding required?

b) When the arch space is sold the total volume must be indicated on the sales brochure. Calculate the total volume in m³ to 2 d.p.

a) The area will be 2 rectangles of sides 10m and 8m plus two semi-circles of radius 5m (this comes from 15m - 10m).

Area = 2 x (10 x 8) + p(5)² (2 semi circles = 1 circle)

Area = 160 + 25p

=238.54 m²

b) Volume = Volume of box (10 x 8 x 16) + the volume of half a cylinder ( half of p(5)²  x  16)

= 1,280 + ¹/₂(p x 25 x 16)

= 1,908.32 m³

4. The diagram shows a small hot water cylinder, which is made up from a cylinder which is capped with a hemisphere. Calculate the maximum volume of water that the cylinder can hold. Give your answer in litres to 3 s.f.

Total Volume = Volume of Cylinder + half of the Volume of a sphere with radius 0.25m.

= p(0.25)²  x  1   +   ½(⁴/₃p(0.25)³)

= 0.229 m³

5. The diagram shows a section of a rocket firework. If this section can be completely filled with gunpowder, what is the volume of gunpowder required?

We first need to use a bit of trigonometry to calculate the radius:

Sin 30^o = r / 6

hence r = 6sin 30^o

r = 3 cm

The total volume = volume of cylinder + volume of cone

= pr²h + ¹/₃pr²h

= pr²(h + ¹/₃h)

= ⁴/₃pr²h

= ⁴/₃p  x  (3)²  x  20

= 240p

= 753.98 cm³

6. A wheelchair ramp is to be constructed at the local town hall with the dimensions as shown in the diagram. The ramp must be made of solid concrete. What volume of concrete is required to make the ramp?

Total volume = volume of cuboid + volume of tirngular prism

Volume of cuboid = 0.5 x 2 x 2 = 2m³

Volume of prism = cross-sectional area  x  length

The cross sectional area is the area of the triangle which is 1/2  x  base  x  height

we know the base is 0.5m but we need to calculate the height - use trig. again for this:

tan 30^o = 0.5 / height

hence height = 0.5 / tan 30^o

= 0.866m (to 3 d.p.)

Area of triangle = 1/2 x 0.5 x 0.866 = 0.217 (to 3 d.p.)

Volume = Area x 2 = 0.217 x 2 = 0.433m³

Hence total Volume = 2 + 0.433 = 2.433 m³
(to 3 d.p.)

7.  The "Peardrop" is part of a silver necklace, its dimensions are shown in the diagram. What volume of silver is required to make the "Peardrop"?

The Peardrop is actually part of a thin cylinder (it is a sector of the cylinder to be precise). If we had a full circle the Volume would be = Volume of a cylinder

= p(25)²  x  5

= 3125p

The easiest way to do this is think of what fraction of a circle 360^o, the angle 40^o is.

40     =  1
360       9

Hence the Volume is 1/9  x 3125p

= 1,090.83 mm³

8. The diagram shows the planned design of a washing machine button. The button comprises of a hemisphere attached to a cuboid with relative dimensions as shown in the diagram.

a) Show that the volume of the button is:

px³ + x²y
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b) Calculate the volume when x = 2cm and y = 1cm.

a) Volume = volume of base + volume of half a sphere

Volume of base = x  x  x  x  y = x²y

Volume of whole sphere with radius = 1/2x is:

⁴/₃p(1/2x)³ 

= ⁴/₃p  x  1/8x³ 

= 1/6px³ 

But we only have half a sphere so the volume is:

1/12px³ 

Hence the total volume = px³ + x²y
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b) Substituting the values for x and y we get:

8p + 4   =  6.094 cm³      (to 3 d.p.)
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