π cm 3 Find the radius of the sphere 1 ii When the sphere is fully immersed in a cylinder of water the level of the water rises by 2 cm Find the radius of the cylinder 2 A candle is in the shape of a cylinder with a conical top as shown in the diagram ID: 1044687
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2. 1.(i) The volume of a sphere is 288π cm3. Find the radius of the sphere.
3. 1.(ii) When the sphere is fully immersed in a cylinder of water, the level of the water rises by 2 cm. Find the radius of the cylinder.
4. 2.A candle is in the shape of a cylinder with a conical top, as shown in the diagram. The cone has height 24 cm and the length of the radius of its base is 10 cm. (i)Find the volume of the cone in terms of π.
5. 2.A candle is in the shape of a cylinder with a conical top, as shown in the diagram. The height of the cylinder is equal to the slant length of the cone. Find the volume of the cylinder in terms of π.(ii)Find the slant length of the cone: Therefore, the height of the cylinder = 26 cm
6. 2.A candle is in the shape of a cylinder with a conical top, as shown in the diagram. The height of the cylinder is equal to the slant length of the cone. Find the volume of the cylinder in terms of π.(ii)Volume of cylinder = πr2h= π(10)2(26) = 2600π cm3
7. 2.A candle is in the shape of a cylinder with a conical top, as shown in the diagram. A solid spherical ball of wax was used to make the candle.Find the radius of this spherical ball, correct to one decimal place.(iii)
8. 2.A candle is in the shape of a cylinder with a conical top, as shown in the diagram. A solid spherical ball of wax was used to make the candle.Find the radius of this spherical ball, correct to one decimal place.(iii)
9. 3.(i) A solid cylinder, made of wax, has a radius of length 15 cm and height of 135 cm. Find its volume in terms of π.Volume of cylinder = πr2h = π(15)2(135) = 30,375π cm3
10. 3.(ii) The solid cylinder is melted down and recast to make four identical right circular solid cones. The height of each cone is equal to twice the length of its base radius. Calculate the base radius length of the cones.
11. 4.A steel-works buys steel in the form of solid cylindrical rods of diameter 20 centimetres and length 30 metres. The steel rods are melted to produce solid spherical ball bearings. No steel is wasted in the process. (i)Find the volume of steel in one cylindrical rod, in terms of π. Diameter = 20 cm, so radius = 10 cm, Height = 30 × 100 = 3000 cm Volume of cylinder = πr2h= π(10)2(3000) = 300,000π cm3
12. 4.A steel-works buys steel in the form of solid cylindrical rods of diameter 20 centimetres and length 30 metres. The steel rods are melted to produce solid spherical ball bearings. No steel is wasted in the process. (ii)The radius of a ball bearing is 2 cm. How many such ball bearings are made from one steel rod?
13. 4.A steel-works buys steel in the form of solid cylindrical rods of diameter 20 centimetres and length 30 metres. The steel rods are melted to produce solid spherical ball bearings. No steel is wasted in the process. (ii)The radius of a ball bearing is 2 cm. How many such ball bearings are made from one steel rod?
14. 4.A steel-works buys steel in the form of solid cylindrical rods of diameter 20 centimetres and length 30 metres. The steel rods are melted to produce solid spherical ball bearings. No steel is wasted in the process. (iii)Ball bearings of a different size are also produced. One steel rod makes 225,000 of these new ball bearings. Find the radius of the new ball bearings.
15. 4.A steel-works buys steel in the form of solid cylindrical rods of diameter 20 centimetres and length 30 metres. The steel rods are melted to produce solid spherical ball bearings. No steel is wasted in the process. (iii)Ball bearings of a different size are also produced. One steel rod makes 225,000 of these new ball bearings. Find the radius of the new ball bearings.
16. 5.(i) Find the volume of a solid sphere with a diameter of length 3 cm. Give your answer in terms of π.Diameter = 3 cm, so radius = 1·5 cm
17. 5.A cylindrical vessel with internal diameter of length 15 cm contains water. The surface of the water is 11 cm from the top of the vessel. How many solid spheres, each with diameter of length 3 cm, must be placed in the vessel in order to bring the surface of the water to 1 cm from the top of the vessel? Assume that all the spheres are submerged in the water.Diameter = 15 cm, so radius = 7·5 cm Height of displaced water = 11 – 1= 10 cm
18. 5.A cylindrical vessel with internal diameter of length 15 cm contains water. The surface of the water is 11 cm from the top of the vessel. How many solid spheres, each with diameter of length 3 cm, must be placed in the vessel in order to bring the surface of the water to 1 cm from the top of the vessel? Assume that all the spheres are submerged in the water.
19. 6.A solid is in the shape of a hemisphere with a cone on top, as in the diagram. The volume of the hemisphere is 18π cm3. Find the radius of the hemisphere. (i)Volume of hemisphere = 18π cm3
20. 6.A solid is in the shape of a hemisphere with a cone on top, as in the diagram. The slant length of the cone is cm.(ii)Show that the vertical height of the cone is 6 cm. Use Pythagoras’ Theorem:
21. 6.A solid is in the shape of a hemisphere with a cone on top, as in the diagram. Show that the volume of the cone equals the volume of the hemisphere.(iii)This is the same as the volume of the hemisphere.
22. 6.A solid is in the shape of a hemisphere with a cone on top, as in the diagram. This solid is melted down and recast in the shape of a solid cylinder.(iv)The height of the cylinder is 9 cm. Calculate its radius. Volume of cylinder = Volume of cone + Volume of hemisphere= 18π + 18π = 36π cm3
23. 6.A solid is in the shape of a hemisphere with a cone on top, as in the diagram. This solid is melted down and recast in the shape of a solid cylinder.(iv)Volume of cylinder = πr2h
24. 7.A solid wax candle is in the shape of a cylinder with a cone on top, as shown in the diagram. The diameter of the base of the cylinder is 6 cm and the height of the cylinder is 12 cm. The volume of the wax in the candle is 123π cm3.Find the total height of the candle.(i)Diameter = 6 cm, so radius = 3 cmVolume of cylinder = πr2h= π(3)2(12) = 108π cm3
25. 7.A solid wax candle is in the shape of a cylinder with a cone on top, as shown in the diagram. The diameter of the base of the cylinder is 6 cm and the height of the cylinder is 12 cm. The volume of the wax in the candle is 123π cm3.Find the total height of the candle.(i)Total height = 12 + 5= 17 cm
26. 7.A solid wax candle is in the shape of a cylinder with a cone on top, as shown in the diagram. The diameter of the base of the cylinder is 6 cm and the height of the cylinder is 12 cm. The volume of the wax in the candle is 123π cm3.20 of these candles fit into a rectangular box. The candles are arranged in five rows of four. Find the dimensions of the smallest rectangular box that the candles will fit into. (ii)Length = 5 × 6 cm = 30 cm Width = 4 × 6 cm = 24 cm Height = 17 cm
27. 7.A solid wax candle is in the shape of a cylinder with a cone on top, as shown in the diagram. The diameter of the base of the cylinder is 6 cm and the height of the cylinder is 12 cm. The volume of the wax in the candle is 123π cm3.When the 20 candles are packed into the box, what percentage of the box is air? (iii)Volume of rectangular box = L × W × H = 30 × 24 × 17 = 12,240 cm3 Volume of 20 candles = 20 × 123π = 7728·32 cm3 Volume of empty space (Air) = Volume of box – Volume of candles= 12,240 – 7728·32 = 4511·68 cm3
28. 7.A solid wax candle is in the shape of a cylinder with a cone on top, as shown in the diagram. The diameter of the base of the cylinder is 6 cm and the height of the cylinder is 12 cm. The volume of the wax in the candle is 123π cm3.When the 20 candles are packed into the box, what percentage of the box is air? (iii)
29. 8.A test tube consists of a cylinder on top of a hemisphere, as shown in the diagram. Find the capacity (internal volume) of the test tube, in terms of π. (i)Diameter = 4 cm, so radius = 2 cm Height of cylinder = 14 – 2= 12 cm
30. 8.A test tube consists of a cylinder on top of a hemisphere, as shown in the diagram. Find the capacity (internal volume) of the test tube, in terms of π. (i)Volume of test-tube
31. 8.A test tube consists of a cylinder on top of a hemisphere, as shown in the diagram. Water is added to the test tube, until it is three-quarters full. How far is the surface of the water from the top of the test tube? (ii)If the tube is three-quarters full, then it is one quarter empty.The empty space is in the cylindrical part of the tube.
32. 8.A test tube consists of a cylinder on top of a hemisphere, as shown in the diagram. Water is added to the test tube, until it is three-quarters full. How far is the surface of the water from the top of the test tube? (ii)Find the height of the empty space: Therefore, the surface of the water is from the top of the tube.
33. 9.(i) A solid metal sphere has a diameter of 9 cm. Find the volume of the sphere in terms of π. Diameter = 9 cm, so radius = 4·5 cm
34. 9.(ii) This sphere is melted down and all the metal is used to make a solid shape which consists of a cone on top of a cylinder, as shown in the diagram. The cone and the cylinder both have height 8 cm. The cylinder and the base of the cone both have radius r cm. Calculate r, correct to one decimal place.
35. 10.A cylindrical tank, of base diameter 40 cm, contains a quantity of water. A solid cone, of height 25 cm and base radius 15 cm is dropped into the cylinder, such that it is upright and some of the cone is above the water (as shown in the diagram). If the depth of the water in the tank is now 20 cm, find the original depth of the water before the cone was dropped into the tank.Volume of the cone immersed in the water = Volume of displaced water
36. 10.A cylindrical tank, of base diameter 40 cm, contains a quantity of water. A solid cone, of height 25 cm and base radius 15 cm is dropped into the cylinder, such that it is upright and some of the cone is above the water (as shown in the diagram). If the depth of the water in the tank is now 20 cm, find the original depth of the water before the cone was dropped into the tank.Find volume of cone inside water: Similar triangles:
37. 10.A cylindrical tank, of base diameter 40 cm, contains a quantity of water. A solid cone, of height 25 cm and base radius 15 cm is dropped into the cylinder, such that it is upright and some of the cone is above the water (as shown in the diagram). If the depth of the water in the tank is now 20 cm, find the original depth of the water before the cone was dropped into the tank.
38. 10.A cylindrical tank, of base diameter 40 cm, contains a quantity of water. A solid cone, of height 25 cm and base radius 15 cm is dropped into the cylinder, such that it is upright and some of the cone is above the water (as shown in the diagram). If the depth of the water in the tank is now 20 cm, find the original depth of the water before the cone was dropped into the tank.
39. 10.A cylindrical tank, of base diameter 40 cm, contains a quantity of water. A solid cone, of height 25 cm and base radius 15 cm is dropped into the cylinder, such that it is upright and some of the cone is above the water (as shown in the diagram). If the depth of the water in the tank is now 20 cm, find the original depth of the water before the cone was dropped into the tank.Therefore, original depth = 20 – 4·65 = 15·35 cm