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Exercise 24 - (c)

Mixed Problems Based on Three Dimensional Solids including Cone and Sphere.

Q. 1. A toy is in the form of a cone mounted on a hemisphere of radius 3.5 cm. If the total height of the toy is 15.5 cm, find its total surface area. (Use π = 22/7).

Q. 2. A toy is in the form of a cone mounted on a hemisphere of diameter 7 cm. The total height of the toy is 14.5 cm. Find the volume and the total surface area of the toy. (Use π = 22/7).

Q. 3. A cone is 8.4 cm high and the radius of its base is 2.1 cm. It is melted and recast into a sphere. Find the radius of the sphere. (Use π = 22/7).

Q. 4. A cone of height 15 cm and diameter 7 cm is mounted on a hemisphere of same diameter. Determine the volume of the solid thus formed. (Use π = 22/7).

Q. 5. A vessel is in the form of a inverted cone. Its height is 8 cm and the radius of its top, which is open is 5 cm. It is filled with water up to the rim. When lead shots each of which is a sphere of radius 0.5 cm are dropped in to the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

Q. 6. 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 up to the rim. When lead shots, each of which is a sphere  of radius 0.25 cm are dropped in to the vessel, 2/5 of the water flows out. Find the number of lead shots dropped into the vessel.

Q. 7. A buoy is made in the form of a hemisphere surmounted by a right circular cone whose base coincides exactly with the plane surface of hemisphere. The radius of the base of the cone is 3.5 m and its volume is 2/3 of the hemisphere. Calculate the height of the cone and the surface area of the buoy, correct to two places of decimal.

Q. 8. 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 cones submerged ?

Q. 9. 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.

Q. 10. 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.

Q. 11. 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 :

1. The radius of the solid sphere.
2. The number of cones recast. (Take π = 3.14).

Q. 12. A girl fills a cylindrical bucket 32 cm in height and 18 cm in radius with sand. She empties the bucket on the ground and makes a conical heapof the sand. If the height of the conical heap is 24 cm. Find :

1. Its radius and
2. Its slant height. (Leave your answer in square root form).

Q. 13. An exhibition tent is in the form of a cylinder surmounted by a cone. The height of the tent above the ground is 85 m and the height of the cylindrical part is 50 m. If the diameter of the base is 168 m, find the quantity of canvas required to make the tent. Allow 20% extra for folds and for stitching. Give your answer to the nearest m².

Q. 14. A metal container in the form of a cylinder is surmounted by a hemisphere of the same radius. The internal height of the cylinder is 7 m and the internal radius is 3.5 m. Calculate :

1. The total area of internal surface, excluding the base;
   The internal volume of the container in m³.(Take π = 22/7)

Q. 15. A solid consisting of right circular cone, standing on a hemisphere, is placed upright in a right circular cylinder, full of water, and touches the bottom. Find the volume of water left in the cylinder, given that the radius of the cylinder is 3 cm and its height is 6 cm; The radius of the hemisphere is 2 cm and the height of the cone is 4 cm. Give your answer to the nearest cubic centimeter. (Take π = 22/7).

Q. 16. A vessel is in the form of a hemispherical bowl mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the capacity of the vessel. (Take π = 22/7).

Q. 17.  A solid is composed of a cylinder with hemispherical ends. If the whole length of the solid is 104 cm and radius of each hemispherical ends is 7 cm, find the cost of polishing its surface at the rate of Rs. 10 per dm².

Q. 18. 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 3 cm. If the height of the toy is 12 cm, find the number of toys so formed.

Q. 19. Water is flowing at the rate of 15 km per hour through a pipe of diameter 14 cm into a rectangular tank which is 50 m long and 44 m wide. Find the time in which the level of water in the tank will rise by 21 cm.

Q. 20. 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. [Use π = 22/7]. 

Q. 21. The rain water from a roof 22 m × 20 m drains into a cylindrical vessel having diameter of base 2 m and height 3.5 m. If the vessel is just full, find the rainfall in cm.

Q. 22. A tent is in the form of a cylinder of diameter 4.2 m and height 4 m, surmounted by a cone of equal base and height 2.8 m. Find the capacity of the tent and the cost of canvas for making the tent at Rs. 100 per sq. m.s

Q. 23. A well of diameter 3m, is dug 14 m deep. The earth taken out of it has been spread evenly all around it to a width of 4 m, to form an embankment. Find the height of the embankment.(Use π = 22/7).

Q. 24. A circus tent is cylindrical to a height of 3 m and conical above it. If its base radius is 52.5 m and slant height of the conical portion is 53 m, find the area of the canvas needed to make the tent. ( Use p = 22/7).