**Exercise 17.1**

**Take π = 22/7 unless stated otherwise.**

**1. Find the total surface area of a solid cylinder of radius 5 cm and height 10 cm. Leave your answer in terms of π.**

**Solution:**

^{2}

^{2}.

**2. An electric geyser is cylindrical in shape, having a diameter of 35 cm and height 1.2m. Neglecting the thickness of its walls, calculate**

**(i) its outer lateral surface area,**

**(ii) its capacity in litres.**

**Solution:**

^{2}

^{2}

^{2}h

^{2}×120

^{3}

^{3}= 1 litre]

^{2}and 115.5 litres respectively.

**3. A school provides milk to the students daily in cylindrical glasses of diameter 7 cm. If the glass is filled with milk upto a height of 12 cm, find how many litres of milk is needed to serve 1600 students.**

**Solution:**

^{2}h

^{2}×12

^{3}

^{3}

^{3}= 1 litre]

**4. In the given figure, a rectangular tin foil of size 22 cm by 16 cm is wrapped around to form a cylinder of height 16 cm. Find the volume of the cylinder.**

**Solution:**

^{2}h

^{2}×16

^{3}

^{3}.

**5. (i) How many cubic metres of soil must be dug out to make a well 20 metres deep and 2 metres in diameter?**

**(ii) If the inner curved surface of the well in part (i) above is to be plastered at the rate of Rs 50 per m**

^{2}, find the cost of plastering.**Solution:**

^{2}h

^{2}×20

^{3}

^{3}.

^{2}

^{2 }= Rs.50

**6. A road roller (in the shape of a cylinder) has a diameter 0.7 m and its width is 1.2 m. Find the least number of revolutions that the roller must make in order to level a playground of size 120 m by 44 m.**

**Solution:**

^{2}

^{2}

**7. If the volume of a cylinder of height 7 cm is 448 π cm**

^{3}, find its lateral surface area and total surface area.**Solution:**

^{3}

^{2}h = 448

^{2}×7 = 448

^{2}= 448/7 = 64

^{2}

^{2}

^{2}and 240 cm

^{2}.

**8. A wooden pole is 7 m high and 20 cm in diameter. Find its weight if the wood weighs 225 kg per m**

^{3}.**Solution:**

^{2}h

^{2}×7

^{3}

^{3}= 225 kg

^{3}wood = 225×0.22 = 49.5 kg

**9. The area of the curved surface of a cylinder is 4400 cm**

^{2}, and the circumference of its base is 110 cm. Find

**(i) the height of the cylinder.**

**(ii) the volume of the cylinder.**

**Solution:**

^{2}

^{2}h

^{2}×40

^{3}

^{3}.

**10. A cylinder has a diameter of 20 cm. The area of curved surface is 1000 cm**

^{2}. Find**(i) the height of the cylinder correct to one decimal place.**

**(ii) the volume of the cylinder correct to one decimal place. (Take π = 3.14)**

**Solution:**

^{2}

^{2}h

^{2}×15.9

^{3}

^{3}

**11. The barrel of a fountain pen, cylindrical in shape, is 7 cm long and 5 mm in diameter. A full barrel of ink in the pen will be used up when writing 310 words on an average. How many words would use up a bottle of ink containing one-fifth of a litre?**

**Answer correct to the nearest. 100 words.**

**Solution:**

^{2}h

^{2}×7

^{3}

**12. Find the ratio between the total surface area of a cylinder to its curved surface area given that its height and radius are 7.5 cm and 3.5 cm.**

**Solution:**

**13. The radius of the base of a right circular cylinder is halved and the height is doubled. What is the ratio of the volume of the new cylinder to that of the original cylinder?**

**Solution:**

_{1}= r

^{2}h

_{2}= r

^{2}h

^{2}×2h

^{2}h

_{2}/V

_{1}= ½ r

^{2}h/r

^{2}h = ½

**14. (i) The sum of the radius and the height of a cylinder is 37 cm and the total surface area of the cylinder is 1628**

**cm**

^{2}**. Find the height and the volume of the cylinder.**

**(ii) The total surface area of a cylinder is 352 cm**

^{2}. If its height is 10 cm, then find the diameter of the base.**Solution:**

^{2}

^{2}h

^{2}×30

^{3}

^{3}respectively.

^{2}

^{2}+10r = (352×7)/2×22

^{2}+10r = 56

^{2}+10r-56 = 0

**15.The ratio between the curved surface and the total surface of a cylinder is 1∶ 2. Find the volume of the cylinder, given that its total surface area is 616 cm**

^{2}.**Solution:**

^{2}

^{2}

^{2}= 616

^{2}= 616

^{2}= 616-308

^{2}= 308

^{2}= 308/2 = 154

^{2}= 154/ = 154×7/22 = 49

^{2}h

^{2}×7

^{3}

^{3}.

**16. Two cylindrical jars contain the same amount of milk. If their diameters are in the ratio 3∶ 4, find the ratio of their heights.**

**Solution:**

_{1}and r

_{2}be the radius of the two cylinders and h

_{1}and h

_{2}be their heights.

_{1}:r

_{2}= 3:4

_{1}

^{2}h

_{1}= r

_{2}

^{2}h

_{2}

_{1}/h

_{2}= r

_{2}

^{2}/ r

_{1}

^{2}

_{1}/h

_{2}= 4

^{2}/3

^{2}= 16/9

**17. A rectangular sheet of tin foil of size 30 cm × 18 cm can be rolled to form a cylinder in two ways along length and along breadth. Find the ratio of volumes of the two cylinders thus formed.**

**Solution:**

_{1}= r

^{2}h

^{2}×18

_{2}= r

^{2}h

^{2}×30

_{1}/V

_{2}= (15×105×9/11)÷( 9×63×15/11)

**19. The given figure shows a metal pipe 77 cm long. The inner diameter of a cross-section is 4 cm and the outer one is 4.4 cm. Find its**

**(i) inner curved surface area**

**(ii) outer curved surface area**

**(iii) total surface area.**

**Solution:**

^{2}

^{2}.

^{2}

^{2}.

^{2}-r

^{2})

^{2}-2

^{2})

^{2}

^{2}.

**20. A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior. The diameter of the pencil is 7 mm and the diameter of the graphite is 1 mm. If the length of the pencil is 14 cm, find the volume of the wood and that of the graphite.**

**Solution:**

^{2}h

^{2}×14

^{3}

^{3}

^{2}-r

^{2})h

^{2}-(1/20)

^{2}]14

^{3}

^{3}

**21. A soft drink is available in two packs**

**(i) a tin can with a rectangular base of length 5 cm and width 4 cm, having a height of 15 cm and**

**(ii) a plastic cylinder with circular base of diameter 7 cm and height 10 cm.**

**Which container has greater capacity and by how much?**

**Solution:**

^{3}

^{2}h

^{2}×10

^{3}

^{3}

^{3}more volume than the tin can.

**22. A cylindrical roller made of iron is 2 m long. Its inner diameter is 35 cm and the thickness is 7 cm all round. Find the weight of the roller in kg, if 1 cm**

^{3}of iron weighs 8 g.**Solution:**

^{2}-r

^{2})h

^{2}-(35/2)

^{2}]200

^{2}-35

^{2}/4]200

^{2}-35

^{2})

^{2}-35

^{2})

**Exercise 17.2**

**Take π = 22/7 unless stated otherwise.**

**1. Write whether the following statements are true or false. Justify your answer.**

**(i) If the radius of a right circular cone is halved and its height is doubled, the volume will remain unchanged.**

**(ii) A cylinder and a right circular cone are having the same base radius and same height. The volume of the cylinder is three times the volume of the cone.**

**(iii) In a right circular cone, height, radius and slant height are always the sides of a right triangle.**

**Solution:**

^{2}h

^{2}2h = (1/3)r

^{2}h/2

^{2}h

^{2}h

^{2}= h

^{2}+r

^{2}

**2. Find the curved surface area of a right circular cone whose slant height is 10 cm and base radius is 7 cm.**

**Solution:**

^{2}

^{2}.

**3. Diameter of the base of a cone is 10.5 cm and slant height is 10 cm. Find its curved surface area.**

**Solution:**

^{2}

^{2}.

**4. Curved surface area of a cone is 308 cm**

^{2}and its slant height is 14 cm. Find ,**(i)radius of the base**

**(ii)total surface area of the cone.**

**Solution:**

^{2}

^{2}+ rl

^{2}+308

^{2}.

**5. Find the volume of the right circular cone with**

**(i) radius 6 cm and height 7 cm**

**(ii) radius 3.5 cm and height 12 cm.**

**Solution:**

^{2}h

^{2}×7

^{3}

^{3}.

^{2}h

^{2}×12

^{3}

^{3}.

**6. Find the capacity in litres of a conical vessel with**

**(i) radius 7 cm, slant height 25 cm**

**(ii) height 12 cm, slant height 13 cm**

**Solution:**

^{2}= h

^{2}+r

^{2}

^{2}h

^{2}×24

^{3}

^{3}]

^{2}= h

^{2}+r

^{2}

**7. A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kiloliters ?**

**Solution:**

^{2}h

^{2}×12

^{3}

^{3}]

**8. If the volume of a right circular cone of height 9 cm is 48π cm**

^{3}, find the diameter of its base.**Solution:**

^{2}h = 48

^{2}×9 = 48

^{2}= 48

^{2}= 48/3 = 16

**9. The height of a cone is 15 cm. If its volume is 1570 cm**

^{2}, find the radius of the base. (Use π = 3.14)**Solution:**

^{3}

^{2}h = 1570

^{2}×15 = 1570

^{2}= 1570

^{2}= 1570/5×3.14 = 314/3.14 = 100

**10. The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white washing its curved surface area at the rate of Rs 210 per 100 m**

^{2}**.**

**Solution:**

^{2}

^{2}.

^{2}= Rs.210

**11. A conical tent is 10 m high and the radius of its base is 24 m. Find :**

**(i) slant height of the tent.**

**(ii) cost of the canvas required to make the tent, if the cost of 1 m**

^{2}canvas is Rs 70.**Solution:**

^{2}= h

^{2}+r

^{2}

^{2}= 10

^{2}+24

^{2}

^{2}= 100+576

^{2}= 676

**12. A Jocker’s cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the cloth required to make 10 such caps.**

**Solution:**

^{2}= h

^{2}+r

^{2}

^{2}= 24

^{2}+7

^{2}

^{2}= 576+49

^{2}= 625

^{2}

^{2}.

**13. (a) The ratio of the base radii of two right circular cones of the same height is 3∶ 4. Find the ratio of their volumes.**

**(b) The ratio of the heights of two right circular cones is 5∶ 2 and that of their base radii is 2∶ 5. Find the ratio of their volumes.**

**(c) The height and the radius of the base of a right circular cone is half the corresponding height and radius of another bigger cone. Find:**

**(i) the ratio of their volumes.**

**(ii) the ratio of their lateral surface areas.**

**Solution:**

_{1}and r

_{2}be the radius of the given cones and h be their height.

_{1}:r

_{2}= 3:4

_{1}= (1/3)r

^{2}h

_{2 }= (1/3)(r/2)

^{2}(h/2) = (1/3)r

^{2}h/8

_{2}/V

_{1}= ( 1/3)r

^{2}h/8÷ (1/3)r

^{2}h

^{2}h ×(3/r

^{2}h)

**14. Find what length of canvas 2 m in width is required to make a conical tent 20 m in diameter and 42 m in slant height allowing 10% for folds and the stitching. Also find the cost of the canvas at the rate of Rs 80 per metre.**

**Solution:**

^{2}

^{2}.

^{2}

**15. The perimeter of the base of a cone is 44 cm and the slant height is 25 cm. Find the volume and the curved surface of the cone.**

**Solution:**

**16. The volume of a right circular cone is 9856 cm**

^{3}and the area of its base is 616 cm^{2}. Find**(i) the slant height of the cone.**

**(ii) total surface area of the cone.**

**Solution:**

^{2}

^{2}= 616

^{2}= 616

^{2}= 616×7/22

^{2}= 196

^{3}

^{2}h = 9856

^{2}×h = 9856

^{2})

**17. A right triangle with sides 6 cm,8 cm and 10 cm is revolved about the side 8 cm. Find the volume and the curved surface of the cone so formed. (Take π = 3.14)**

**Solution:**

^{2}h

^{3}

^{3}.

^{2}.

**18. The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to its base. If its volume be 1/27 of the volume of the given cone, at what height above the base is the section cut?**

**Solution:**

^{2}H

^{2}h = (1/27)× (1/3)R

^{2}H

^{2}h = (1/27)× (1/3)R

^{2}×30

^{2}h/R

^{2}= 30/27

^{2}h/R

^{2}= 10/9 ….(i)

^{2}×h = 10/9

^{3}/900 = 10/9

^{3}= 900×10/9 = 1000

**19. A semi-circular lamina of radius 35 cm is folded so that the two bounding radii are joined together to form a cone. Find**

**(i) the radius of the cone.**

**(ii) the (lateral) surface area of the cone.**

**Solution:**

_{1}be radius of cone.

_{1}

_{1}

_{1}

_{1}= 35/2= 17.5 cm

_{1}l

^{2}

_{2}.

**Exercise 17.3**

**1. Find the surface area of a sphere of radius :**

**(i) 14 cm**

**(ii) 10.5 cm**

**Solution:**

^{2}

^{2}

^{3}

^{2}.

^{2}

^{2}

^{3}

^{2}.

**2. Find the volume of a sphere of radius :**

**(i) 0.63 m**

**(ii) 11.2 cm**

**Solution:**

^{3}

^{3}

^{3}

^{3}(approx)

^{3}.

^{3}

^{3}

^{3}

^{3}(approx)

^{3}.

**3. Find the surface area of a sphere of diameter:**

**(i) 21 cm**

**(ii) 3.5 cm**

**Solution:**

^{2}

^{2}

^{2}

^{2}.

^{2}

^{2}

^{2}

^{2}.

**4. A shot-put is a metallic sphere of radius 4.9 cm. If the density of the metal is 7.8 g per cm**

^{3}, find the mass of the shot-put.**Solution:**

^{3}

^{3 }

^{3}

**5. Find the diameter of a sphere whose surface area is 154 cm**

^{2}.**Solution:**

^{2}

^{2}

^{2}= 154

^{2}= 154 ×7/(22×4)

**6. Find:**

**(i) the curved surface area.**

**(ii) the total surface area of a hemisphere of radius 21 cm.**

**Solution:**

^{2}

^{2}

^{2}

^{2}.

^{2}

^{2}

^{2}

^{2}.

**7. A hemispherical brass bowl has inner- diameter 10.5 cm. Find the cost of tin-plating it on the inside at the rate of Rs 16 per 100 cm**

^{2}.**Solution:**

^{2}

^{2}

^{2}

^{2}

**8. The radius of a spherical balloon increases from 7 cm to 14 cm as air is jumped into it. Find the ratio of the surface areas of the balloon in two cases.**

**Solution:**

^{2}

^{2}/4R

^{2}

^{2}/R2

^{2}/14

^{2}

**9. A sphere and a cube have the same surface area. Show that the ratio of the volume of the sphere to that of the cube is √6 ∶√π**

**Solution:**

^{2}

^{2}

**10. (a) If the ratio of the radii of two sphere is 3∶ 7, find :**

**(i) the ratio of their volumes.**

**(ii) the ratio of their surface areas.**

**(b) If the ratio of the volumes of the two sphere is 125∶ 64, find the ratio of their surface areas.**

**Solution:**

_{1}and r

_{2}.

^{3}

**11. A cube of side 4 cm contains a sphere touching its sides. Find the volume of the gap in between.**

**Solution:**

^{3}

^{3}

^{3}

^{3}

^{3}(approx)

^{3}(approx)

^{3}.

**12. Find the volume of a sphere whose surface area is 154 cm**

^{2}.**Solution:**

^{2}

^{2}= 154

^{2}= 154

^{2}= (154×7)/(4×22)

^{2}= 49/4

^{3}

^{3}

^{3}(approx)

^{3}

**14. A hemispherical bowl has a radius of 3.5 cm. What would be the volume of water it would contain?**

**Solution:**

^{3}

^{3}

**15. The water for a factory is stored in a hemispherical tank whose internal diameter is 14 m. The tank contains 50 kilolitres of water. Water is pumped into the tank to fill to its capacity. Find the volume of water pumped into the tank.**

**Solution:**

^{3}

^{3}

^{3}

^{3}(approx)

**16. The surface area of a solid sphere is 1256 cm². It is cut into two hemispheres. Find the total surface area and the volume of a hemisphere. Take π = 3.14.**

**Solution:**

^{2}

^{2}= 1256

^{2}= 1256

^{2}= 1256×/3.14×4

^{2}= 100

^{2}

^{2}

^{2}

^{2}.

^{3}

^{3}

**17. Write whether the following statements are true or false. Justify your answer :**

**(i) The volume of a sphere is equal to two-third of the volume of a cylinder whose height and diameter are equal to the diameter of the sphere.**

**(ii) The volume of the largest right circular cone that can be fitted in a cube whose edge is 2r equals the volume of a hemisphere of radius r.**

**(iii) A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is 1∶ 2∶ 3.**

**Solution:**

^{2}h

^{2}×2r

^{3}

^{3}

^{3}

^{2}h

^{2}×2r

^{3}

^{2}h

^{3}

^{3}

^{2}h

^{3 }

^{3}: (2/3)r

^{3}: r

^{3}

**Exercise 17.4**

**Take π = 22/7 unless stated otherwise.**

**1. The adjoining figure shows a cuboidal block of wood through which a circular cylindrical hole of the biggest size is drilled. Find the volume of the wood left in the block.**

**Solution:**

^{3}

^{2}h

^{2}×70

^{3}

^{3}

^{3}.

**2. The given figure shows a solid trophy made of shining glass. If one cubic centimetre of glass costs Rs 0.75, find the cost of the glass for making the trophy.**

**Solution:**

^{3}

^{3}

^{3}

^{2}h

^{2}×28

^{3}

^{3}

^{3}glass = Rs.0.75

**3. From a cube of edge 14 cm, a cone of maximum size is carved out. Find the volume of the remaining material.**

**Solution:**

^{3}

^{3}

^{3}

^{2}h

^{2}×14

^{3}

**4. A cone of maximum volume is curved out of a block of wood of size 20 cm x 10 cm x 10 cm. Find the volume of the remaining wood.**

**Solution:**

^{3}[Volume = lbh]

^{2}h

^{2}×20

^{3}

^{3}

^{3}.

**5. 16 glass spheres each of radius 2 cm are packed in a cuboidal box of internal dimensions 16 cm × 8 cm × 8 cm and then the box is filled with water. Find the volume of the water filled in the box.**

**Solution:**

**6. A pen stand 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 depression is 0.5 cm and the depth is 1.4 cm. Find the volume of the wood in the entire stand, correct to 2 decimal places.**

**Solution:**

^{3}

^{2}h

^{2}×1.4

^{3}

^{3}

^{3}

^{3}.

**7. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter that the hemisphere can have? Also, find the surface area of the solid.**

**Solution:**

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}.

**8. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder (as shown in the given figure). If the height of the cylinder is 10 cm and its base is of radius 3.5 cm, find the total surface area of the article.**

**Solution:**

^{2}

^{2}

^{2}.

**9. A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. If the total height of the toy is 15.5 cm, find the total surface area of the toy.**

**Solution:**

**10. A circus tent is in the shape of a cylinder surmounted by a cone. The diameter of the cylindrical portion is 24 m and its height is 11 m. If the vertex of the cone is 16 m above the ground, find the area of the canvas used to make the tent.**

**Solution:**

**Solution:**

**12. From a solid cylinder of height 30 cm and radius 7 cm, a conical cavity of height 24 cm and of base radius 7 cm is drilled out. Find the volume and the total surface of the remaining solid.**

**Solution:**

^{2}H – (1/3)r

^{2}h

^{2}(H- h/3)

^{3}

^{3}.

^{2}+ rl

^{2}

^{2}.

**13. The adjoining figure shows a wooden toy rocket which is in the shape of a circular cone mounted on a circular cylinder. The total height of the rocket is 26 cm, while the height of the conical part is 6 cm. The base of the conical portion has a diameter of 5 cm, while the base diameter of the cylindrical portion is 3 cm. If the conical portion is to be painted green and the cylindrical portion red, find the area of the rocket painted with each of these colours.**

**Also, find the volume of the wood in the rocket. Use π = 3.14 and give answers correct to 2 decimal places.**

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}.

^{2}

^{2}

^{2}.

^{2}H + r

^{2}h

^{2}H/3) + r

^{2}h))

^{2}×6/3) + 1.5

^{2}×20)

^{3}

^{3}

**14. The adjoining figure shows a hemisphere of radius 5 cm surmounted by a right circular cone of base radius 5 cm. Find the volume of the solid if the height of the cone is 7 cm. Give your answer correct to two places of decimal.**

**Solution:**

^{3}+ (1/3)r

^{2}h

^{2}(2r+h)

^{2}(2×5+7)

^{3}

^{3}

^{3}.

**15. A buoy is made in the form of a hemisphere surmounted by a right cone whose circular base coincides with the plane surface of the hemisphere. The radius of the base of the cone is 3.5 metres and its volume is 2/3 of the hemisphere. Calculate the height of the cone and the surface area of the buoy correct to 2 places of decimal.**

**Solution:**

^{3}

^{3}

^{3}

^{3}

^{2}h

^{2}h = 539/9

^{2}×h = 539/9

**16. A circular hall (big room) has a hemispherical roof. The greatest height is equal to the inner diameter, find the area of the floor, given that the capacity of the hall is 48510 m**

^{3}.**Solution:**

^{3 }

^{3 }+ (2/3)r

^{3 }[∵h = r]

^{3 }(1+2/3)

^{3 }(3+2)/3

^{3 }

^{3}

^{3 }= 48510

^{3 }= 48510

^{3 }= 48510×3×7/(22×5)

^{3 }= 9261

^{2}

^{2}

^{2}

^{2}.

**17. A building is in the form of a cylinder surmounted by a hemisphere valted dome and contains 41 19/21 m**

^{3}of air. If the internal diameter of dome is equal to its total height above the floor, find the height of the building.**Solution:**

^{2}h + (2/3)r

^{3}

^{3}+ (2/3)r

^{3}[∵h = r]

^{3}(1+2/3)

^{3}(3+2)/3

^{3}

^{3}= 880/21

^{3 }= 880/21

^{3}= 880×3×7/(5×22×21)

^{3}= 880/110

^{3}= 8

**18. A rocket is in the form of a right circular cylinder closed at the lower end and surmounted by a cone with the same radius as that of the cylinder. The diameter and the height of the cylinder are 6 cm and 12 cm respectively. If the slant height of the conical portion is 5 cm, find the total surface area and the volume of the rocket. (Use π = 3.14).**

**Solution:**

^{2}((h/3)+H)

^{3}

^{3}.

**19. The adjoining figure represents a solid consisting of a right circular cylinder with a hemisphere at one end and a cone at the other. Their common radius is 7 cm. The height of the cylinder and the cone are each of 4 cm. Find the volume of the solid.**

**Solution:**

^{2}h +r

^{2}H + (2/3) r

^{3}

^{2}((h/3) + H + (2r/3))

^{2}×((4/3)+4+(2×7)/3)

^{3}

^{3}.

**20. A solid is in the form of a right circular cylinder with a hemisphere at one end and a cone at the other end. Their common diameter is 3.5 cm and the height of the cylindrical and conical portions are 10 cm and 6 cm respectively. Find the volume of the solid. (Take π = 3.14)**

**Solution:**

^{2}h +r

^{2}H + (2/3) r

^{3}

^{2}((h/3) + H + (2r/3))

^{2}×((6/3)+10+(2×1.75)/3)

^{3}

^{3}

^{3}.

**21. 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 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 the height of the conical part is 12 cm.**

**Solution:**

^{2}+rl

^{2}

^{2}.

**22.The adjoining figure shows a model of a solid consisting of a cylinder surmounted by a hemisphere at one end. If the model is drawn to a scale of 1∶ 200, find**

**(i) the total surface area of the solid in π m².**

**(ii) the volume of the solid in π litres.**

**Solution:**

^{2}+2rh + 2r

^{2}

^{2}

^{2}

^{2}.

^{2}h + (2/3) r

^{3}

^{2}(h+ (2/3)r)

^{2}(1600+ (2/3)×600)

^{3}

^{3}

^{3}= 1000 litres]

**Exercise 17.5**

**1. The diameter of a metallic sphere is 6 cm. The sphere is melted and drawn into a wire of uniform cross-section. If the length of the wire is 36 m, find its radius.**

**Solution:**

^{3}

^{3}

^{3}

^{2}h = 36

^{2}×3600 = 36

^{2 }= 1/100

**2. The radius of a sphere is 9 cm. It is melted and drawn into a wire of diameter 2 mm. Find the length of the wire in metres.**

**Solution:**

^{3}

^{3}

^{3}

^{2}h = 972

^{2}×h =972

^{2}

**3. A solid metallic hemisphere of radius 8 cm is melted and recasted into right circular cone of base radius 6 cm. Determine the height of the cone.**

**Solution:**

^{3}

^{3}

^{3}

^{2}h = (1024/3)

^{2}×h = (1024/3)

**4. A rectangular water tank of base 11 m × 6 m contains water upto a height of 5 m. if the water in the tank is transferred to a cylindrical tank of radius 3.5 m, find the height of the water level in the tank.**

**Solution:**

^{3}

^{2}h

^{2}h = 330

^{2}×h = 330

**5. The rain water from a roof of dimensions 22 m × 20 m drains into a cylindrical vessel having diameter of base 2 m and height; 3.5 m. If the rain water collected from the roof just fill the cylindrical vessel, then find the rainfall in cm.**

^{3}

^{2}h

^{2}×3.5

^{3}

**6. The volume of a cone is the same as that of the cylinder whose height is 9 cm and diameter 40 cm. Find the radius of the base of the cone if its height is 108 cm.**

**Solution:**

^{2}h

^{2}×9

^{3}

^{2}h

^{2}×108

^{2}

^{2}= 3600

^{2}= 3600/36

^{2}= 100

**7. Eight metallic spheres, each of radius 2 cm, are melted and cast into a single sphere. Calculate the radius of the new (single) sphere.**

**Solution:**

^{3}

^{3}

^{3}

^{3}

^{3}

^{3}= (256/3)

^{3}= 256

^{3}= 256/4 = 64

**8. A metallic disc, in the shape of a right circular cylinder, is of height 2.5 mm and base radius 12 cm. Metallic disc is melted and made into a sphere. Calculate the radius of the sphere.**

**Solution:**

^{2}h

^{2}×0.25

^{3}

^{3}

^{3 }= 36

^{3}= 36×3/4

^{3}= 27

**9. Two spheres of the 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 big sphere.**

**Solution:**

_{1 }= 1 kg

_{2}= 7 kg

_{3}= 1+7 = 8 kg

_{3}be volume of bigger sphere.

_{1}/V

_{1}= m

_{3}/V

_{3}

_{1}= 8/V3

_{1}/ V

_{3}= 1/8 …(i)

_{1 }= (4/3)r

^{3}

_{1}= (4/3) ××3

^{3}

_{1}= 36

_{1}/ V_3= 36÷(4/3)R

^{3}

_{1}/ V_3= 27/R

^{3}…(ii)

^{3}

^{3}= 27×8 = 216

**10. A hollow copper pipe of inner diameter 6 cm and outer diameter 10 cm is melted and changed into a solid circular cylinder of the same height as that of the pipe. Find the diameter of the solid cylinder.**

**Solution:**

^{2}-r

^{2})h

^{2}-3

^{2})×h

^{3}

^{2}h

^{2}h = 16h

^{2}= 16

**11. A solid sphere of radius 6 cm is melted into a hollow cylinder of uniform thickness. If the external radius of the base of the cylinder is 4 cm and height is 72 cm, find the uniform thickness of the cylinder.**

**Solution:**

^{3}

^{3}

^{3}

^{2}-r

^{2})h

^{2}-r

^{2})h = 288

^{2}-r

^{2})×72 = 288

^{2}-r

^{2}) = 288/72

^{2}-r

^{2}) = 4

^{2}= 4

^{2}= 16-4

^{2}= 12

**12. A hollow metallic cylindrical tube has an internal radius of 3 cm and height 21 cm. The thickness of the metal of the tube is ½ cm. The tube is melted and cast into a right circular cone of height 7 cm. Find the radius of the cone correct to one decimal place.**

**Solution:**

^{2}-r

^{2})h

^{2}-3

^{2})×21

^{3}

^{2}h

^{2}×7

^{2}

^{2}= 68.25

^{2}= 68.25×3/7 = 29.25

**13. A hollow sphere of internal and external diameters 4 cm and 8 cm respectively, is melted into a cone of base diameter 8 cm. Find the height of the cone. (2002)**

**Solution:**

^{3}-r

^{3})

^{3}-2

^{3})

^{2}h

^{2}×h

**14. A well with inner diameter 6 m is dug 22 m deep. Soil taken out of it has been spread evenly all round it to a width of 5 m to form an embankment. Find the height of the embankment.**

**Solution:**

^{2}H

^{2}×22

^{3}

^{2}– r

^{2})h

^{2}– 3

^{2})h

**15. A cylindrical can of internal diameter 21 cm contains water. A solid sphere whose diameter is 10.5 cm is lowered into the cylindrical can. The sphere is completely immersed in water. Calculate the rise in water level, assuming that no water overflows.**

**Solution:**

^{2}h = (4/3)r

^{3}

^{2}h = (4/3)××(21/4)

^{3}

**16. There is water to a height of 14 cm in a cylindrical glass jar of radius 8 cm. Inside the water there is a sphere of diameter 12 cm completely immersed. By what height will the water go down when the sphere is removed?**

**Solution:**

^{2}h = (4/3)r

^{3}

^{2}h = (4/3)6^3

**17. A vessel in the form of an inverted cone is filled with water to the brim. Its height is 20 cm and diameter is 16.8 cm. Two equal solid cones are dropped in it so that they are fully submerged. As a result, one-third of the water in the original cone overflows. What is the volume of each of the solid cone submerged?**

**Solution:**

^{2}h

^{2}×20

^{3}

^{3}

^{3}

^{3}

^{3}.

**18. A solid metallic circular cylinder of radius 14 cm and height 12 cm is melted and recast into small cubes of edge 2 cm. How many such cubes can be made from the solid cylinder?**

**Solution:**

^{2}h

^{2}×12

^{3}

^{3}

^{3}= 8 cm

^{3}

**19. How many shots each having diameter 3 cm can be made from a cuboidal lead solid of dimensions 9 cm × 11 cm × 12 cm?**

**Solution:**

^{3}

^{3}

^{3}

^{3}

^{3}

**20. How many spherical lead shots of diameter 4 cm can be made out of a solid cube of lead whose edge measures 44 cm?**

**Solution:**

^{3}

^{3 }

^{3 }

^{3}

^{3}

^{3}

**21. Find the number of metallic circular discs with 1.5 cm base diameter and height 0.2 cm to be melted to form a circular cylinder of height 10 cm and diameter 4.5 cm.**

**Solution:**

^{2}h

^{2}×10 = 50.625 cm

^{3}

^{2}h

^{2}×0.2

^{3}

**22. A solid metal cylinder of radius 14 cm and height 21 cm is melted down and recast into spheres of radius 3.5 cm. Calculate the number of spheres that can be made.**

**Solution:**

^{2}h

^{2}×21

^{3}

^{3}

^{3}

^{3}

**23. A metallic sphere of radius 10.5 cm is melted and then recast into small cones, each of radius 3.5 cm and height 3 cm. Find the number of cones thus obtained.**

**Solution:**

^{3}

^{3}

^{3}

^{2}h

^{2}×3

^{3}

**24. A certain number of metallic cones each of radius 2 cm and height 3 cm are melted and recast in a solid sphere of radius 6 cm. Find the number of cones.**

**Solution:**

^{2}h

^{2}×3

^{3}

^{3}

^{3}

^{3}

**25. A vessel is in the form of an inverted cone. Its height is 11 cm and the radius of its top, which is open, is 2.5 cm. It is filled with water upto the rim. When some lead shots, each of which is a sphere of radius 0.25 cm, are dropped into the vessel, 2/5 of the water flows out. Find the number of lead shots dropped into the vessel.**

**Solution:**

^{2}h

^{2}×11

^{3}

^{3}

^{3}

^{3}

^{3}

**26. The surface area of a solid metallic sphere is 616 cm². It is melted and recast into smaller spheres of diameter 3.5 cm. How many such spheres can be obtained?**

**Solution:**

^{2}

^{2}= 616

^{2}= 616

^{2}= 616×7/4×22

^{2}= 49

^{3}

^{3}

^{3}

^{3}

^{3}

^{3}

**27. The surface area of a solid metallic sphere is 1256 cm². It is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate (i) the radius of the solid sphere. (ii) the number of cones recast. (Use π = 3.14).**

**Solution:**

^{2}

^{2}= 1256

^{2}= 1256

^{2 }= 1256/4×3.14

^{2 }= 100

^{3}

^{3}

^{3}

^{3}

^{2}h

**28. Water is flowing at the rate of 15 km/h through a pipe of diameter 14 cm into a cuboid pond which is 50 m long and 44 m wide. In what time will the level of water in the pond rise by 21 cm?**

**Solution:**

^{3 }

^{2}h

^{2}×h

**29. A cylindrical can whose base is horizontal and of radius 3.5 cm contains sufficient water so that when a sphere is placed in the can, the water just covers the sphere. Given that the sphere just fits into the can, calculate : (i) the total surface area of the can in contact with water when the sphere is in it. (ii) the depth of the water in the can before the sphere was put into the can. Given your answer as proper fractions.**

**Solution:**

^{2}

^{2}

^{2}.

^{2}h = (4/3)r

^{3}+r

^{2}d

^{2}h = r

^{2}{(4/3)r +d)}