Surface Areas and Volume
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1 Surface Areas and Volume 1. How many cubes of length 2cm can be made from a sphere of lead of radius 14cm? 2. A hemispherical bowl of internal diameter 30cm contains some liquid. This liquid is to be filled into cylindrical shaped bottles each of diameter 5cm and height 6cm. Find the number of bottles necessary to empty the bowl circular pates, each of radius 7cm and thickness 1 cm, are placed 2 one above the other to form a solid right circular cylinder. Find the total surface area and volume of the cylinder so formed. 4. A bucket made up of a metal sheet is in the form of a frustum of a cone of height 16cm with diameters of its lower and upper ends as 16cm and 40cm respectively. Find the volume of the bucket. Also find the cost of the bucket if the cost of the metal sheet used is Rs. 15 per 100cm². 5. The diameter of the internal external surfaces of a hollow spherical shell are 6cm and 10 cm respectively. If it is melted and recast into a solid cylinder of diameter 14cm, find the height of the cylinder. 6. A conical vessel whose internal radii is 5 cm and height 24 cm is full of water. The water is emptied into a cylindrical vessel with internal radius 10cm. Find the height to which the water rises in the cylindrical vessel. 7. A cone of height 24cm and radius of base 6cm is made up of modeling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere.
2 8. An agricultural field is in the form of a rectangle of length 20 m and width 14m. A 10 m deep well of diameter 7m is dug in a corner of the well and the earth taken out of the well is spread evenly over the remaining part of the field. Find the rise in its level. 9. The radii of the internal and external surfaces of a metallic spherical shell are 3 cm and 5cm respectively. It is melted and recast into a solid right circular cylinder of height 10 2 cm. Find the diameter of the base of the 3 cylinder. 10. A sphere of diameter 5cm is dropped into a vessel partly filled with water. The diameter of the vessel is 10cm. If the sphere is completely submerged, by how much will the level of water rise? persons took dip in a rectangular tank which is 80m long and 50m broad. What is the rise in level water in the tank if the average displacement of water by a person is 0.04m³? 12. A hemispherical tank of radius 1 m is full of water. It is connected with a pipe which empties it at the rate of 7 litres per second. How much time will it take to empty the tank completely. 13. A circus tent is made of canvas and is in the shape of a right circular cylinder and a right circular cone above it. Thediameter and height of the cylindrical part of the tent are 126m and 5m respectively. The total height of the tent is 21m. Find the total cost of the tent if the canvas used costs Rs 12 persquare meter. 14. A toy is the form of a cone mounted on a hemisphere of diameter 7cm. The total height of the toy is 14.5cm. Find the volume and the total surface area of the toy. (Taken π = 22 7 )
3 15. A right triangle whose sides are 15 cm and 20cm is made to revolve about its hypotenuse. Find the volume and the surface area of the double cone thus formed. (Taken π = 22 7 ) 16. A container made up of a metal sheet is in the form of a frustum of a cone of height 16cm with radii of its lower and upper ends as 8cm and 20 cm respectively. Find the cost of milk which can completely fill the container at the rate of Rs. 15 per litre and the cost of the metal sheet used if it costs Rs 5 per 100cm². (Taken π = 22 7 ) 17. Find the volume of a solid in the form of a right circular cylinder with hemispherical end whose total length is 2.7 m and the diameter of each hemispherical end is 0.7m. 18. A petrol tank is a cylinder of base diameter 21 cm and length 18 cm fitted with conical ends each of axis length 9cm. Determine the capacity of the tank. 19. From a solid right circular cylinder with height 10cm and radius of base 6cm, a right circular cone of the same height and same base is removed. Find the volume of the reamining solid. 20. A cone of radius 10cm is divided into two parts by drawing a plane through the midpoint of its axis, parallel to its base. Compare the volume of the two parts. 21. The diameter of a sphere is 42cm. It is melted and drawn into a cylindrical wire of 28cm diameter. Find the length of the wire. 22. A spherical object of radius 14 cm is dropped into water contained in a right circular cylindrical vessel of radius 21cm. If the object is completely immersed, find how much level of the water is raised? 23. The surface area of a sphere is 616 cm². Find its radius
4 24. The length of two cylinders are in the ratio 5:2 and their diameters are in the ratio 1:2. Calculate the ratio of their volumes. 25. If the radius of the sphere is made twice what will be its new surface area? 26. The curved surface area of a cylinder 2772cm² and its base radius is 21cm, find the height of the cylinder. 27. Three cubes each of side 5cm are joined end to end. Find the surface area of the resulting cuboid 28. The diameter of a garden roller is 1.4 m and it is 2m long. How much area will it cover in 5 revolutions? 29. The area of a base of a cone is 616cm². If its height is 48cm, find its total surface area. 30. A cylinder, a cone and a hemi-sphere have equal base and same height. What is the ratio of their volume? 31. The surface area and volume of a sphere are numerically equal. Find the diameter. 32. The total surface areas of a solid sphere and a solid hemisphere are equal, what will be the ratio of their radii? 33. The radius of the Conical tent is 5 feet and its top is 12 feet high from the ground. Find the slant height of the tent. 34. Find the surface area of a cricket ball of radius 2 inches. 35. A toy is in the form of a cone mounted on a hemisphere of common base radius 7cm. The total height of the toy is 31 cm find the total surface area of the toy (use π = 22 7 ) 36. A sphere of diameter 12cm, is dropped in a right circular cylindrical vessel, partly filled with water. If the sphere is completely submerged in water, the water level in the cylindrical vessel rises by 3 5 cm. Find the diameter of the cylindrical vessel. 9
5 37. The radius of the base and height of a right circular cylinder are in the ratio 2:3 and its volume is 1617cm³. Find the total surface area of the cylinder. (use π = 22 7 ) 38. A solid metalic sphere of diameter 21 cm is melted and recast into a number of smaller cones, each of diameter 7 cm and height 3cm. Find the number of cones so formed. 39. A hemispherical bowl of internal diameter 30cm contains some liquid. This liquid is to be filled into cylindrical shaped bottles each of diameter 5cm and height 6cm. Find the number of bottles necessary to empty the bowl. 40. Water is flowing at the rate of 7m per second through a circular pipe whose internal diameter is 2 cm into a cylindrical tank the radius of whose base is 40 cm. Determine the increase in the water in ½ hour 41. What length of a rectangular cloth 42 cm wide will be required to make a conical cap of height 28 cm and base radius 21 cm? The length of material will be required for stitiching margins and wastage in cutting is 10 cm. 42. Find the area of the metal sheet required to make an open tank in the shape of a cone of height 24 m and diameter 14m. 43. Three cones, of same height and same radius are melted and recast into a single sphere of radius 12cm. Find the height of the cone if its diameter is 12 cm. 44. A cylindrical tube open at one end is made of metal. The internal diameter of the tube is 20 cm and its length is 21 cm. The metal every where is 4 cm in thickness. Calculate the volume of the metal. 45. A solid metalic sphere of diameter 28 cm is melted and recast into a number of smaller cones, each of diameter 42 3 so formed cm and height 3cm. Find the number of cones
6 46. A solid sphere of radius 3 cm is melted and then cast into small spherical balls each of diameter 0.6 cm. Find the number of balls thus obtained 47. A sphere of diameter 6 cm is dropped in a right circular cylindrical vessel partly filled with water. The diameter of that vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel? 48. The diameter of a copper sphere is 18 cm. The spere is melted and is drawn into a long wire of uniform circular cross section. If the length of the wire is 108m, find its diameter 49. If the radius of the circular ends of a bucket which is 45cm high, are 28cm and 7cm, find the capacity of the bucket. 50. A conical vessel whose internal radius is 5cm and height 24 cm is full of water. The water is emptied into a cylindrical vessel with internal radius 10cm. Find the height to which the water rises. 51. From the top of a cone of base radius 21 cm and height 72 cm, a cone of slant height 25 cm is cut off. What is the volume and curved surface area of the remaining portion? 52. Find the area of a metal sheet required to make an open tank in the shape of a frustum of height 24m, lower diameter 14m and upper radius 14m. 53. The internal and external radii of a hollow sphere are 3 cm and 5cm respectively. The sphere is melted to form a solid cylinder of height 2 2 cm. Find the diameter 3 and curved surface area of the cylinder. 54. A vessel in the shape of an inverted cone of height 72cm 42cm and diameter 42 cm is filled with milk at a height of 48cm. Find the surface area of the vessel, which is not in contact 72cm 48cm with the milk. Also find the volume of the vessel.
7 55. Water flows out through a circular pipe, whose internal diameter is 2cm, at the rate of 0.7m per second into a cylindrical tank. The radius of whose base is 40cm. By how much will the level of water in cylindrical tank rise in half an hour? 56. A bucket is in the form of a frustum of a cone with a capacity of cm³ of water. The radii of the top and bottom circular ends are 20cm and 12cm respectively. Find the height of the bucket and the area of the metal sheet used its making. (use π = 3.14) 57. A solid cylinder of diameter 12 cm and height 15 cm is melted and recast into toys with the shape of a right circular cone mounted on a hemisphere of radius 3cm. If the height of the toy is 12cm, find the number of toys so formed. 58. A hollow cone is cut by a plane parallel to the base and the upper portion is removed. If the curved surface of the remainder is 8 th of the curved surface of the whole 9 cone, find the ratio of the line segments into which the cone s altitude is divided by the plane. 59. A circus tent is cylindrical to a height of 3m and conical above it. If its base radius is 52.5m and slant height of the conical portion is 53 m, find the area of the canvas needed to make the tent. (use π = 22 7 ) 60. A rocket in the shape of a cylinder with a cone attached to one end and a hemisphere attached to the other. All of them of the same radius. The total length of the rocket is 8m and the cone and cylinder of same height of 3m. Compute the volume of the rocket. 61. The rain water from a roof 22 m x 20m drains into a cylindrical vessel having diameter of base 2m and height 3.5m. If the vessel is just full, find the rainfall in cm.
8 62. A tank is in the shape of a right circular cone. The angle between the height and slant height of it is 30. How much water can fill this tank if its slant height is 42cm. 63. A bucket made up of a metal sheet is in the form of a frustum of a cone. Its depth is 24 cm and the diameters of the top and bottom are 30 cm and 10 cm respectively. Find the cost of milk which can completely fill the bucket at the rate of Rs. 20 per litre and the cost of the metal sheet used, if it costs Rs. 10 per 100cm². (use π = 3.14) 64. A solid right circular cone of diameter 14cm and height 8 cm is melted to form a hollow sphere. If the external diameter of the sphere is 10cm, find the internal diameter of the sphere.
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