Surface Areas and Volumes
1. A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top which is open, is 5 cm. It is filled with water upto the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of water flows out. Find the number of lead shots dropped into the vessel.
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2. How many coins of 1.75 cm in diameter and 2 mm thick must be melted to form a cuboid of 11 cm × 10 cm × 7 cm?
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3. A well is to be dug with 4 m inside diameter and 10 m in depth. Find the quantity of earth to be excavated. The earth taken out is spread all round to a width of 4 m to form an embankment. Find the height of the embankment.
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4. Two cubes each of volume of 64 cm3 are joined end to end. Find the surface area of the resulting cuboid.
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5. Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere?
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6. Total surface area of a cube is 216 cm2. Find its volume.
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7. If a solid right-circular cone of height 24 cm and base radius 6 cm is melted and recast in the shape of a sphere, find the radius of the sphere.
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8. Find the curved surface area of a right-circular cone of height 15 cm and base diameter 16 cm.
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9. Two cones have their heights in the ratio 1 : 3 and radii in the ratio 3 : 1. What is the ratio of their volumes?
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10. 12 solid spheres of the same radii are made by melting a solid metallic cylinder of base diameter 2 cm and height 16 cm. Find the diameter of each sphere.
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11. Three solid metal cubes of edges 6 cm, 8 cm and 10 cm are melted and recasted into a single solid cube. Find the length of the edge of the cube so obtained.
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12. A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of uniform thickness. Find the thickness of the wire.
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13. Two spheres of same metal weigh 1 kg and 7 kg. The radius of the smaller sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the diameter of the new sphere.
View AnswerAns. 12 cm
14. A cone and a cylinder have the same radii but the height of the cone is 3 times that of the cylinder. Find the ratio of their volumes.
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15. A solid sphere is melted and recasted into a hollow cylinder of uniform thickness. If the external radius of the base of the cylinder is 4 cm, its height 24 cm and thickness 2 cm; find the radius of the sphere.
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16. A heap of rice is in the form of a cone of base diameter 24 m and height 3.5 m. Find the volume of the rice. How much canvas cloth is required to just cover the heap?
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17. A cone of height 24 cm and radius of base 6 cm is made up of modelling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere.
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18. How many silver coins, 1.75 cm in diameter and of thickness 2 mm, must be melted to form a cuboid of dimensions 5.5 cm × 10 cm × 3.5 cm?
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19. The 3/4 th part of a conical vessel of internal radius 5 cm and height 24 cm is full of water. The water is emptied into a cylindrical vessel with internal radius 10 cm. Find the height of water in cylindrical vessel.
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20. Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes, if 8 cm standing water is needed.
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21. In a hospital, used water is collected in a cylindrical tank of diameter 2 m and height 5 m. After recycling, this water is used to irrigate a park of hospital whose length is 25 m and breadth is 20 m. If tank is filled completely then what will be the height of standing water used for irrigating the park?
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22. A well, whose diameter is 3 m, has been dug 21 m deep and the earth dug out is used to form an embankment 4 m wide around it. Find the height of the embankment.
View AnswerAns. 1.69 m
23. In a cylindrical vessel of radius 10 cm, containing some water, 9000 small spherical balls are dropped which are completely immersed in water which raises the water level. If each spherical ball is of radius 0.5 cm, then find the rise in the level of water in the vessel.
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24. Water is flowing through a cylindrical pipe of internal diameter 2 cm, into a cylindrical tank of base radius 40 cm at the rate of 0.7 m/sec. By how much will the water rise in the tank in half an hour?
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25. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter hemisphere can have? Find the surface area of the solid.
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26. Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.)
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27. A playing top (lattu) is shaped like a cone surmounted by a hemisphere. The entire top is 5 cm in height and the diameter of top is 3.5 cm. Find its whole surface area.
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28. A pen stand (as shown in the figure) made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand.
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29. A rectangular metal block has length 15 cm, breadth 10 cm and height 5 cm. From this block, a circular hole of diameter 7 cm is drilled out. Find:
(i) the volume of the remaining solid
(ii) the surface area of the remaining solid.
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30. A toy is in the form of a cone mounted on a hemisphere of radius 3.5 cm. The total height of the toy is 15.5 cm. Find the total surface area of the toy.
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31. A solid is in the form of a right circular cylinder with hemispherical ends. The total height of the solid is 58 cm and the diameter of the cylinder is 28 cm. Find the total surface area of the solid. (Take π = 22/7).
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32. A toy is in the shape of a right circular cylinder with a hemisphere on one end and a cone on the other. The height and the radius of the cylindrical part are 13 cm and 5 cm respectively. The radii of the hemispherical and conical parts are the same as that of the cylindrical part. Calculate the surface area of the toy if height of the conical part is 12 cm.
View AnswerAns. 770 cm2
33. A gulab jamun when completely ready for eating contains sugar syrup upto about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns shaped like a cylinder with two hemispherical ends, if the complete length of each of the gulab jamuns is 5 cm and its diameter is 2.8 cm.
View AnswerAns. 337.88 cm3
34. A juice seller was serving his customers using glasses. The inner diameter of the cylindrical glass was 5 cm, but the bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the height of the glass was 10 cm, find its actual capacity and its apparent capacity
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