Name______________________________ Grade___________ Date____________
1
The diagram shows a roll of material.
The material is wound onto a metal
cylinder whose cross-section is a circle
of radius 10 cm.
The shaded area shows the cross-section
of the material on the roll.
The outer layer of material forms the
curved surface of a cylinder of
radius 30 cm.
30
10
200
(i) Calculate, in square centimetres, the area of the cross-section of the material on the roll (shaded
on the diagram).
[2]
2
(ii) The material is 200 cm wide on the roll.
Calculate, in cubic metres, the volume of the material.
[2]
(iii) When unwound, the length of the material is 150 m.
Calculate the thickness of the material, giving your answer in millimetres.
[2]
[The volume of a cone =
1
3
× base area × height.]
The diagram shows a plant pot.
The open end of the plant pot is a circle of radius 10 cm.
The closed end is a circle of radius 5 cm.
The height of the plant pot is 12 cm.
The plant pot is part of a right circular cone of height 24 cm.
10
12
5
12
Calculate the volume of the plant pot.
Give your answer in litres.
[4]
How many of these plant pots can be completely filled from a 75 litre bag of compost?
[2]
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Name______________________________ Grade___________ Date____________
3
[Surface area of a sphere = 4πr 2]
4
[Volume of a sphere = π r 3]
3
30
A hot water tank is made by joining
a hemisphere of radius 30 cm to an
open cylinder of radius 30 cm and
height 70 cm.
70
30
(a) Calculate the total surface area, including the base, of the outside of the tank.
[4]
(b) The tank is full of water.
(i) Calculate the number of litres of water in the tank.
[3]
(ii) The water drains from the tank at a rate of 3 litres per second.
Calculate the time, in minutes and seconds, to empty the tank.
4
[2]
(a) When a solid rectangular wooden block of oak floats, 60% of its height is under water.
(i) What fraction of its height is above water?
[1]
(ii) A block of oak has length 60 cm,
breadth 50 cm and height
h centimetres.
It floats with 15 cm of its height
under water.
(a) Find the value of h.
15
h
50
60
[1]
(b) In the diagram, the shaded region represents part of the surface area of the block that is in
contact with the water.
Calculate the total surface area of the block that is in contact with the water.
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[2]
Name______________________________ Grade___________ Date____________
5
[The surface area of a sphere is 4πr 2.]
[The volume of a cone is
1
3
× base area × height.]
[The area of the curved surface of a cone of radius r and slant height l is πrl.]
4
16
3
A drinking glass consists of a hollow cone attached to a solid hemispherical base as shown in the
diagram.
The hemisphere has a radius of 3 cm.
The radius of the top of the cone is 4 cm and the height of the cone is 16 cm.
(a) Calculate the total surface area of the solid hemispherical base.
[3]
(b) Calculate the curved surface area of the outside of the cone.
[3]
(c) (i) The cone contains liquid to a depth of d centimetres.
Giving your reasons, show that the radius of the surface of the liquid is 1 d centimetres. [1]
4
(ii) The cone is completely filled with liquid.
Calculate the volume of the liquid.
[2]
(iii) Half of the volume of the liquid from the full cone is now poured out.
Using the answers to parts (i) and (ii), find the depth of the liquid that remains in the cone.
[3]
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Name______________________________ Grade___________ Date____________
6
A, B, C, D and E are five different shaped blocks of ice stored in a refrigerated room.
(a) At 11 p.m. on Monday the cooling system failed, and the blocks started to melt.
At the end of each 24 hour period, the volume of each block was 12% less than its volume at the start
of that period.
(i) Block A had a volume of 7500 cm3 at 11 p.m. on Monday.
Calculate its volume at 11 p.m. on Wednesday.
[2]
(ii) Block B had a volume of 6490 cm3 at 11 p.m. on Tuesday.
Calculate its volume at 11 p.m. on the previous day.
[2]
(iii) Showing your working clearly, find on which day the volume of Block C was half its volume at
11 p.m. on Monday.
[2]
(b) [The volume of a sphere is 43πr 3.]
[The surface area of a sphere is 4πr 2.]
At 11 p.m. on Monday Block D was a hemisphere with radius 18 cm.
Calculate
(i) its volume,
[2]
(ii) its total surface area.
[2]
(c) As Block E melted, its shape was always geometrically similar to its original shape.
It had a volume of 5000 cm3 when its height was 12 cm.
Calculate its height when its volume was 1080 cm3.
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[2]