7.5 Conversion of Solids from one shape to another with no change in Volume

Conversions or Transformations becomes a common part of our daily life. For example, a gold smith melts a bar of gold to transform it to a jewel. Similarly, a kid playing with clay shapes it into different toys, a carpenter uses the wooden logs to form different house hold articles/furniture. Likewise, the conversion of solids from one shape to another is required for various purposes.

In this section we will be learning problems involving conversions of solids from one shape to another with no change in volume. Example . A metallic sphere of radius cm is melted and recast into small spheres each of radius cm. How many small spheres can be obtained?

Solution Let the number of small spheres obtained be n . Let r be the radius of each small sphere and R be the radius of metallic sphere. Here, R = cm, r = cm Now, n ´ (Volume of a small sphere) = Volume of big metallic sphere  = p R n = n = ⇒ n = Therefore, there will be small spheres. Example .

A cone of height cm is made up of modeling clay. A child reshapes it in the form of a cylinder of same radius as cone. Find the height of the cylinder. Solution Let h and h be the heights of a cone and cylinder respectively.

Also, let r be the raius of the cone. Given that, height of the cone h cm; radius of the cone and cylinder r = cm Since, Volume of cylinder = Volume of cone p r h = p r h h = ´ h ⇒ h = Therefore, height of cylinder is cm Example . A right circular cylindrical container of base radius cm and height cm is full of ice cream. The ice cream is to be filled in cones of height cm and base radius cm, having a hemispherical cap.

Find the number of cones needed to empty the container. Solution Let h and r be the height and radius of the cylinder respectively. Given that, h = cm, r = cm Volume of the container V r h = p cubic units. × × × Let, r = cm, h = cm be the radius and height of the cone.

Also, r = cm is the radius of the hemispherical cap. Volume of one ice cream cone = (Volume of the cone + Volume of the hemispherical cap) r h × × × × × × ) = Number of cones = volume of the cylinder volume of oneice cream cone Number of ice cream cones needed = × × × = Thus ice cream cones are required to empty the cylindrical container. Activity The adjacent figure shows a cylindrical can with two balls. The can is just large enough so that two balls will fit inside with the lid on.

The radius of each tennis ball is cm. Calculate the following (i) height of the cylinder. (ii) radius of the cylinder. (iii) volume of the cylinder.

(iv) volume of two balls. (v) volume of the cylinder not occupied by the balls. (vi) percentage of the volume occupied by the balls. Exercise .

. An aluminium sphere of radius cm is melted to make a cylinder of radius cm. Find the height of the cylinder. .

Water is flowing at the rate of km per hour through a pipe of diameter cm into a rectangular tank which is m long and m wide. Find the time in which the level of water in the tanks will rise by cm. . A conical flask is full of water.

The flask has base radius r units and height h units, the water is poured into a cylindrical flask of base radius xr units. Find the height of water in the cylindrical flask. Fig. .

Mensuration . A solid right circular cone of diameter cm and height cm is melted to form a hollow sphere. If the external diameter of the sphere is cm, find the internal diameter. .

Seenu’s house has an overhead tank in the shape of a cylinder. This is filled by pumping water from a sump (underground tank) which is in the shape of a cuboid. The sump has dimensions m ´ . m ´ m.

The overhead tank has its radius of cm and height cm. Find the volume of the water left in the sump after the overhead tank has been completely filled with water from the sump which has been full, initially. . The internal and external diameter of a hollow hemispherical shell are cm and cm respectively.

If it is melted and recast into a solid cylinder of diameter cm, then find the height of the cylinder. . A solid sphere of radius cm is melted into a hollow cylinder of uniform thickness. If the external radius of the base of the cylinder is cm and its height is cm, then find the thickness of the cylinder.

. A hemispherical bowl is filled to the brim with juice. The juice is poured into a cylindrical vessel whose radius is % more than its height. If the diameter is same for both the bowl and the cylinder then find the percentage of juice that can be transferred from the bowl into the cylindrical vessel.

Exercise . Multiple choice questions . The curved surface area of a right circular cone of height cm and base diameter cm is ( A ) p cm ( B ) p cm ( C ) p cm ( D ) p cm . If two solid hemispheres of same base radius r units are joined together along their bases, then curved surface area of this new solid is