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Class 10: Cone and Sphere – ICSE Board Problems

Notes: Important formulas to be kept in mind:

Parameters of a Cone: Radius of the base (), Height of the cone () and Slant Height of a Cone ()

Volume of a Cone

Curved Surface area of a Cone

Total Surface area of a Cone

Volume of a Sphere

Surface area of a Sphere


Question 1: The volume of a conical tent is and the area of the base floor is . Calculate the: (i) radius of the floor (ii) height of the tent (iii) length of the canvas required to cover this conical tent if its width is .    [2008]

Answer:

Volume

Area of the base

(i)

(ii) 

(iii) Curved Surface Area

Length of canvas needed

Question 2: A solid sphere of radius is melted and recast into solid right circular cones of radius and height . Calculate the number of cones recast.   [2013]

Answer:

Sphere: Radius

Cone: Radius and Height

Therefore number of cones re-casted

Question 3: A hollow sphere of internal and external diameters and respectively is melted into a cone of base diameter . Find the height of the cone.   [2002]

Answer:

Internal diameter Internal radius

External diameter External radius

Radius of the

Question 4: A hemispherical bowl of diameter is filled completely with chocolate sauce. This sauce is poured into an inverted cone of radius . Find the height of the cone if it is completely filled.    [2010]

Answer:

Hemisphere: Radius

Cone: Radius , Height

Question 5: A solid cone of radius and height is melted and made into small spheres of radius . Find the number of spheres formed.    [2011]

Answer:

Cone: Radius , Height

Sphere: Radius

Question 6: The total area of a solid metallic sphere is . It is melted and recast into solid right circular cones of radius and height . Calculate: (i) the radius of the solid sphere, (ii) the number of cones recast.   [2000]

Answer:

Surface area

(i)

(ii) Cone: Radius , Height

Question 7: A hollow sphere of internal and external and respectively is melted and recast into small cones of base radius and height . Find the number of cones.    [2012]

Answer:

Sphere: Internal radius , External radius

Cone: Radius , Height

Question 8: The surface area of a solid metallic sphere is . It is melted and recast into solid right circular cones of radius and height . Calculate: (i) the radius of the sphere (ii) the number of cones recast. (Take   [2014]

Answer:

Surface area of sphere

Cone: Radius , Height

(i)

Hence

(ii) 

Question 9: A vessel in the form of an inverted cone, is filled with water to the brim. Its height is and diameter is . Two equal solid cone are dropped in it so that they are fully submerged. As a result, one third of the water in original cone overflows. What is the volume of each of the solid cones submerged?     [2006]

Answer:

Conical Vessel: Height , Radius

The amount of water overflow

Therefore the volume of one sphere

Question 10: A metallic sphere of radius is melted and then recast into small cones each of radius and height . Find the number of cones thus formed.    [2005]

Answer:

Volume of the sphere melted

Volume of the cones formed

Therefore the number of cones formed

Question 11: A girl fills a cylindrical bucket in height and in radius with sand. She empties the bucket on the ground and makes a conical heap of the sand. If the height of the conical heap is , find:  (i) the radius and (ii) the slant height of the heap. Give your answer correct to one place of decimal.    [2004]

Answer:

Volume of the conical heap = Volume of the bucket

(i)

(ii)

Question 12: A vessel is in the form of inverted cone. Its height is and the radius of its top, which is open, is . It is filled with water to the rim. When lead shots, each of which is a sphere of radius , are dropped into the vessel, of the water flows out. Find the number of lead shots dropped into the vessel.    [2003]

Answer:

Volume of water in the cone

Volume of the lead shots = volume of the water that over flows

Volume of one lead shot

Therefore the number of lead shots

Question 13: A solid, consisting of a 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, having given that the radius of the cylinder is and its height is ; the radius of the hemisphere is and the height of cone is . Give your answer to the nearest cubic centimeter.    [1998]

Answer:

Cylinder: Radius , Height

Cone: Radius , Height

Hemisphere: Radius

Volume of cylinder

Volume of hemisphere

Volume of cone

Therefore the volume of water left in the cylinder

Question 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  and the internal radius is . Calculate: (i) the total area of the internal surface, excluding the base;  (ii) the internal volume of the container in   [1999]

Answer:

Cylinder: Radius , Height

Hemisphere: Radius

(i) Internal surface area of cylinder

Internal surface are of hemisphere

Therefore the total surface area

(ii) Internal volume = Volume of the cylinder + volume of the hemisphere

Question 15: An exhibition tent is in the form of a cylinder surmounted by a cone. The height of the tent above the ground is and the height of the cylindrical part is . If the diameter of the base is , find the quantity of the canvas required to make the tent. Allow extra for folds and for stitching. Give your answer to the nearest .    [2001] 

Answer:

Cylinder: Radius , Height

Cone: Radius , Height

Surface area of the cylinder

Curved surface area of the cone

Total surface area

Total cloth required (including 20%)

Question 16:  An open cylindrical vessel of internal diameter and height stands on a horizontal table. Inside this is placed a solid metallic right circular cone, the diameter of whose base is and height is . Find the volume of the water required to fill the vessel. If this cone is replaced by another cone whose height is and the radius of whose base is , find the drop in the water level.    [1993]

Answer:

Cylinder: Radius , Height

Cone: Radius , Height

(i) Volume of cylinder

Volume of the cone

Therefore the volume of the water

(ii) New Cone: Radius

Volume of new cone

Therefore change in volume

Change in height:

Question 17: A cylindrical can, whose base is horizontal and of radius , 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 is contact with water when the sphere is in it (ii) the depth of water in the can before the sphere was put into it.    [1997] 

Answer:

Cylinder: Radius , Height

Hemisphere: Radius

(i) Surface Area

(ii) Volume of sphere

Decrease in height:

Original height



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Class 10: Cone and Sphere – ICSE Board Problems

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