ASSIGNMENT Surface Areas and Volumes

Worksheet Name: 10th Surface Standard: 10th Subject: Mathematics Areas & Volumes, Worksheet ti it if    ​    flti fi lf ti        itif    i tit tifiti fl i t         iflf Q1. How many cubic cen metres of iron is required to construct an open box whose external dimensions are 36cm, 25cm and 16.5cm provided the thickness of the iron is 1.5cm. If one cubic cm of iron weighs 7.5g, nd the weight of the box. Q2. Marbles of diameter 1.4cm are dropped into a cylindrical beaker of diameter 7cm containing some water. Find the number of marbles that should be dropped into the beaker so that the water level rises by 5.6cm. Q3. A solid rectangular block of dimensions 4.4m, 2.6m and 1m is cast into a hollow cylindrical pipe of internal radius 30cm and thickness 5cm. Find the length of the pipe. Q4. A farmer connects a pipe of internal diameter 20cm from a canal into a cylindrical tank which is 10m in diameter and 2m deep. If the water ows through the pipe at the rate of 4km/hr, then in how much me will the tank be lled completely? Q5. Water ows through a cylindrical pipe, whose inner radius is 1cm, at the rate of 80cm/ sec in an empty cylindrical tank, the radius of whose base is 40cm. What is the rise of water level in tank in half an hour? Q6. Three cubes of a metal whose edges are in the ra o 3 : 4 : 5 are melted and converted into a single cube whose diagonal is 12√3cm Find the edges of the three cubes. Q7. A wall 24m long, 0.4m thick and 6m high is constructed with the bricks each of dimensions 25cm × 16cm × 10cm. If the mortar occupies 1 th of the volume of the wall, then nd the number of bricks used in construc ng the wall. 10 Q8. Two solid cones A and B are placed in a cylinderical tube as shown in the Fig. The ra o of their capaci es are 2 : 1. Find the heights and capaci es of cones. Also, nd the volume of the remaining por on of the cylinder. Q9. Water is owing through a cylindrical pipe of internal diameter 2cm, into a cylindrical tank of base radius 40cm, at the rate of 0.4m per second. Determine the rise in level of water in the tank in half an hour. Q10. A hollow sphere of external and internal diameters 8cm and 4cm, respec vely is melted into a solid cone of base diameter 8cm. Find the height of the cone. Q11. Two cones with same base radius 8cm and height 15cm are joined together along their bases. Find the surface area of the shape so formed. Q12. An iron spherical ball has been melted and recast into smaller balls of equal size. If the radius of each of the smaller balls is 1 of 4 the radius of the original ball, how many such balls are made? Compare the surface area, of all the smaller balls combined together with that of the original ball. Q13. A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains 41 19 3 of air. If the internal m 21 diameter of dome is equal to its total height above the oor, nd the height of the building? Q14. The barrel of a fountain pen, cylindrical in shape, is 7cm long and 5mm in diameter. A full barrel of ink in the pen is used up on wri ng 3300 words on an average. How many words can be wri en in a bo le of ink containing one h of a litre? Q15. A right triangle whose sides are 15cm and 20cm (other than hypotenuse), is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed. (Choose value of π as found appropriate) Q16. Water ows at the rate of 10m/ minute through a cylindrical pipe 5mm in diameter. How long would it take to ll a conical vessel whose diameter at the base is 40cm and depth 24cm? Q17. From a cubical piece of wood of side 21cm, a hemisphere is carved out in such a way that the diameter of the hemisphere is equal to the side of the cubical piece. Find the surface area and volume of the remaining piece. Q18. A farmer connects a pipe of internal diameter 25cm from a canal into a cylindrical tank in his eld, which is 12m in diameter and 2.5m deep. If water ows through the pipe at the rate of 3.6km/hr, then in how much me Will the tank be lled? Also, nd the cost of water if the canal department charges at the rate of ₹ 0.07 per m3. Q19. 50 spherical marbles, each of diameter 1.4 cm, are dropped in a cylindrical vessel of diameter 7 cm containing some water, which are completely immersed in water. Find the rise in the level of water in the vessel. Q20. Due to heavy oods in a state, thousands were rendered homeless. 50 schools collec vely o ered to the state government to 1

provide place and the canvas for 1500 tents to be xed by the government and decided to share the whole expenditure equally. The lower part of each tent is cylindrical of base radius 2.8 m and height 3.5 m, with conical upper part of same base radius but of height 2.1 m. If the canvas used to make the tents costs 120 per sqm, nd the amount shared by each school to set up the tents. What value is generated by the above problem? (Use π = 22 ) 7 Q21. The radius of the base of a right circular cone of semi-ver cal angle α is r. Show that its volume is 2 cot  a and curved surface πr area is 2 a. πr cosec Q22. The rain water from a roof of 44m × 20m drains into a cylindrical tank having diameter of base 4m and height 3.5m. If the tank is just full, then nd the rainfall in cm. Q23. Water in a canal, 5.4m wide and 1.8m deep, is owing with a speed of 25km/hr. How much area can it irriggate in 40 minutes, if 10cm of standing water is required for irriga on? Q24. In a rain-water harves ng system, the rain-water from a roof of 22 m × 20 m drains into a cylindrical tank having diameter of base 2 m and height 3.5 m. If the tank is full, nd the rainfall in cm. Write your views on water conserva on. Q25. A toy is in the form of a hemisphere surmounted by a right circular cone of the same base radius as that of the hemisphere. If the radius of the base of the cone is 21cm and its volume is 2 of the volume of hemisphere, calculate the height of the cone 3 and the surface area of the toy. Q26. A well of diameter 2m is dug 14m deep. The earth taken out of it is spread evenly all around it to form an embankment of height 40cm. Find the width of the embankment. Q27. A right circular cylinder and a right circular cone have equal bases and equal heights. If their curved surfaces are in the ra o 8 : 5, determine the ra o of the radius of the base to the height of either of them. Q28. The diameters of the internal and external surfaces of a hollow spherical shell are 6cm and 10cm respec vely. If it is melted and recast into a solid cylinder of diameter 14cm. nd the height of the cylinder. Q29. Sushant has a vessel, of the form of an inverted cone, open at the top, of height 11 cm and radius of top as 2.5 cm and is full of water. Metallic spherical balls each of diameter 0.5 cm are put in the vessel due to which 2 th of the water in the vessel ows 5 out. Find how many balls were put in the vessel. Sushant made the arrangement so that the water that ows out irrigates the ower beds. What value has been shown by Sushant? Q30. A military tent of height 8.25m is in the form of a right circular cylinder of base diameter 30m and height 5.5m surmounted by a right circular cone of same base radius. Find the length of the canvas used in making the tent, if the breadth of the canvas is 1.5m. Q31. 150 spherical marbles, each of diameter 1.4 cm, are dropped in a cylindrical vessel of diameter 7 cm containing some water, which are completely immersed in water. Find the rise in the level of water in the vessel. Q32. Water is owing at the rate of 15km/ h through a pipe of diameter 14cm into a cuboidal pond which is 50m long and 44m wide. In what me will the level of water in pond rise by 21cm? Q33. A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a pla orm 22 m by 14 m. Find the height of the pla orm. Q34. Water is owing through a cylindrical pipe, of internal diameter 2 cm, into a cylindrical tank of base radius 40 cm, at the rate of 0.4 m/s. Determine the rise in level of water in the tank in half an hour. Q35. A metallic cylinder has radius 3cm and height 5cm. To reuce its weight, a conical hole is drilled in the cylinder. The conical hole has a radius of 3 cm and its depth is 8 cm. Calculate the ra o of the volume of metal le in the cylinder to the volume of metal 29 taken out in conical shape. Q36. Water is owing at the rate of 2.52km/h through a cylindrical pipe into a cylindrical tank, the radius of whose base is 40cm, If the increase in the level of water in the tank, in half an hour is 3.15m, nd the internal diameter of the pipe. 2

Q37. 16 glass spheres each of radius 2cm are packed into a cuboidal box of internal dimensions 16cm × 8cm × 8cm and then the box is lled with water. Find the volume of water lled in the box. Q38. Water running in a cylindrical pipe of inner diameter 7cm, is collected in a container at the rate of 192.5 litres per minute. Find the rate of ow of water in the pipe in km/ h.[Use π = 22 ] 7 Q39. Water is owing at the rate of 15km/ hour through a pipe of diameter 14cm into a cuboidal pond which is 50m long and 44m wide. In what me will the level of water in the pond rise by 21cm? Q40. Water is owing at the rate of 6km/hr through a pipe of diameter 14cm into a rectangular tank which is 60m long and 22m wide. Determine the me in which the level of water in the tank Will rise by 7cm. Q41. Marbles of diameter 1.4cm are dropped into a cylindrical beaker of diameter 7cm, containing some water. Find the number of marbles that should be dropped into the beaker so that the water level rises by 5.6cm. Q42. Water ows at the rate of 10m/ minutes through a cylindrical pipe 5mm in diameter. How long would it take to ll a conical vessel whose diameter at the base is 40cm and depth 24cm? Q43. A path 2m wide surrounds a circular pond of diameter 40m. How many cubic metres of gravel are required to grave the path to a depth of 20cm? Q44. A cylindrical tube of radius 12cm contains water to a depth of 20cm. A spherical ball dropped into the tub and the level of the water is raised by 6.75cm. Find the radius of the ball. Q45. In the middle of a rectangular eld measuring 30m × 20m, a well of 7m diameter and 10m depth is dug. The earth so removed is evenly spread over the remaining part of the eld. Find the height through which the level of the eld is raised. Q46. 500 persons are taking a dip into a cuboidal pond which is 80m long and 50m broad. What is the rise of water level in the pond, if the average displacement of the water by a person is 0.04m3? Q47. Three cubes of iron whose edges are 6cm, 8cm and 10cm, respec vely are melted and formed into a single cube. Find the edge of the new cube formed. Q48. Water is owing at the rate of 2.52km/h through a cylindrical pipe into a cylindrical tank, the radius of whose base is 40 cm, If the increase in the level of water in the tank, in half an hour is 3.15m, nd the internal diameter of the pipe. Q49. The 3 th part of a conical vessel of internal radius 5cm and height 24cm is full of water. The water is emp ed into a cylindrical 4 vessel with internal radius 10cm. Find the height of water in cylindrical vessel. Q50. Find the ra o of the volume of a cube to that of a sphere which will t inside it. Q51. A sphere of diameter 12 cm, is dropped in a right circular cylindrical vessel, partly lled 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 Q52. In Fig. 5, from a cuboidal solid metallic block, of dimensions 15cm × 10cm × 5cm a cylindrical hole of diameter 7cm is drilled out. Find the surface area of the remaining block [Use π = 22 ] 7 Q53. A cone, a hemisphere and a cylinder stand on equal bases and have the same height. Show that their volumes are in the ra o 1 : 2 : 3. Q54. The slant height of a conical mountain is 2.5km and the area of its base is 1.54km2. Find the height of the untain. Q55. Sushant has a vessel, of the form of an inverted cone, open at the top, of height 11 cm and radius of top as 2.5 cm and is full of water. Metallic spherical balls each of diameter 0.5 cm are put in the vessel due to which 25th of the water in the vessel ows out. Find how many balls were put in the vessel. Sushant made the arrangement so that the water that ows out irrigates the ower beds. What value has been shown by Sushant? Q56. The internal and external diameters of a hollow hemispherical vessel are 21cm and 25.2cm respec vely. The cost of pain ng 1cm2 of the surface is 10 paise. Find the total cost to paint the vessel all over. Q57. Water is owing at the rate of 2.52km/h through a cylindrical pipe into a cylindrical tank, the radius of whose base is 40cm, If the increase in the level of water in the tank, in half an hour is 3.15m, nd the internal diameter of the pipe. Q58. An icecream cone full of icecream having radius 5cm and height 10cm as shown’in the gure. Calculate the volume of icecream, 3

provided that its 1 parts is le un lled with icecream. 6 Q59. A cylinder, a cone and a hemisphere are of equal base and have the same height. What is the ra o of their volumes? Q60. A sphere and a cube have equal surface areas. What is the ra o of the volume of the sphere to that of the cube? Q61. A copper rod of diameter 1cm and length 8cm is drawn into a wire of length 18m of uniform thickness. Find the thickness of the wire. Q62. Find the ra o of the volumes of a cylinder, a cone and a sphere, if each has the same diameter and same height? Q63. Water in a canal, 5.4 m wide and 1.8 m deep, is owing with a speed of 25 km/hour. How much area can it irrigate in 40 minutes, if 10 cm of standing water is required for irriga on? Q64. What is the ra o of the volume of a cube to that of a sphere which will t inside it? Q65. A wall 24m, 0.4m thick and 6m high is constructed with the bricks each of dimensions 25cm x 16cm x 10cm. If the mortar occupies ( 1 ) of the volume of the wall, then nd the number of bricks used in construc ng the wall. 10 Q66. Prove that the surface area of a sphere is equal to the curved surface area of the circumscribed cylinder. Q67. Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere? Q68. If r1 and r2 be the radii of two solid metallic spheres and if they are melted into one solid sphere, prove that the radius of the new sphere is 3 3 1 (r + r ). 1 23 Q69. Water running in a cylindrical pipe of inneer diameter 7cm, is collected in a container at the rate of 192.5 litres per minute. Find the rate o ow of water in the pipe in km/hr. Q70. A canal is 300cm wide and 120cm deep. The water in the canal is owing with a speed of 20km/h. How much area will it irrigate in 20 minutes if 8cm of standing water is desired? Q71. Sushant has a vessel, of the form of an inverted cone, open at the top, of height 11cm and radius of top as 2.5cm and is full of th water. Metallic spherical balls each of diameter 0.5cm are put in the vessel due to which ( 2 ) of the water in the vessel ows 5 out. Find how many balls were put in the vessel. Sushant made the arrangement so that the water that ows out irrigates the ower beds. What value has been shown by Sushant? Q72. A hemispherical bowl of internal diameter 36 cm contains liquid. This liquid is lled into 72 cylindrical bo les of diameter 6 cm. Find the height of the each bo le, if 10% liquid is wasted in this transfer. Q73. 500 persons are taking dip into a cuboidal pond which is 80m long and 50m broad. What is the rise of water level in the pond, if the average displacement of the water by a person is 0.04m3? Q74. A sphere of maximum volume is cut-out from a solid hemisphere of radius r, what is the ra o of the volume of the hemisphere to that of the cut-out sphere? Q75. In Fig. 5, from a cuboidal solid metallic block, of dimensions 15cm × 10cm × 5cm, a cylindrical hole of diameter 7cm is drilled out. Find the surface area of the remaining block [Use π = 22 ] 7 Q76. Water ows through a cylindrical pipe, whose inner radius is 1cm, at the rate of 80cm/sec in an empty cylindrical tank, the radius of whose base is 40cm. What is the rise of water level in tank in half an hour? Q77. The dimensions of a solid iron cuboid are 4.4 m × 2.6 m × 1.6 m. It is melted and recast into a hollow cylindrical pipe of 30 4

cm inner radius and thickness 5 cm. Find the length of the pipe. Q78. A wire of diameter 3mm is wound about a cylinder whose height is 12cm and radius 5cm so as to cover the curved surface of the cylinder completely. Find the length of the wire. 5