In the previous section, we have discussed how to find the surface area of solids made up of a combination of two basic solids. Here, we shall see how to calculate their volumes.
Volume of combination can be found out by adding volumes of different solids or by subtracting volumes of different solids.
Example 1- Shanta runs an industry in a shed which is in the shape of a cuboid surmounted by a half cylinder. If the base of the shed is of dimension 7 m × 15 m, and the height of the cuboidal portion is 8 m, find the volume of air that the shed can hold. Further, suppose the machinery in the shed occupies a total space of `300 m^3`, and there are 20 workers, each of whom occupy about `0.08 m^3` space on an average. Then, how much air is in the shed? `(Take pi= 22/7)`
Solution- Volume of air in shed= Volume of shed- Space occupied by workers and machinery
Volume of shed= Volume of cuboid+ Volume of half cylinder
`= (8 xx 7 xx 15)+ 1/2 pi r^2h`
= `(8 xx 7 xx 15)+ 1/2 xx 22/7 xx (7/2)^2 xx 15`
Volume of shed =`1128.75 cm^3`
Volume of air in shed= Volume of shed- Space occupied by workers and machinery
`= 1128.75- 300+ (20 xx 0.08)`
Volume of air in shed= `827.15 m^3`
Example 2- A juice seller was serving his customers using glasses. The inner diameter of the cylindrical glass was 5 cm, but the bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the height of a glass was 10 cm, find the apparent capacity of the glass and its actual capacity. (Use π = 3.14.)
Solution: Capacity of the glass= Volume of glass- Volume of hemisphere
= `pi r^2h -2/3 pi r^3`
= `[3.14 (2.5)^2 xx 10] -[2/3 xx 3.14 xx (2.5)^3]`
`"Capacity of the glass"= 163.54 cm^3`
The sum of the inner and the outer curved surfaces of a hollow metallic cylinder is 1056 cm2 and the volume of material in it is 1056 cm3. Find its internal and external radii. Given that the height of the cylinder is 21 cm.
The radius of a solid right circular cylinder increases by 20% and its height decreases by 20%. Find the percentage change in its volume.
The radius of a solid right circular cylinder decreases by 20% and its height increases by 10%. Find the percentage change in its : volume
The radius of a solid right circular cylinder decreases by 20% and its height increases by 10%. Find the percentage change in its : curved surface area.
Find the minimum length in cm and correct to nearest whole number of the thin metal sheet required to make a hollow and closed cylindrical box of diameter 20 cm and height 35 cm.
Given that the width of the metal sheet is 1 m. Also, find the cost of the sheet at the rate of Rs. 56 per m.
Find the area of metal sheet required, if 10% of it is wasted in cutting, overlapping, etc.
A metal cube of side 11 cm is completely submerged in water contained in a cylindrical vessel with diameter 28 cm. Find the rise in the level of water.
The given figure shows a solid formed of a solid cube of side 40cm and a solid cylinder of radius 20 cm and height 50 cm attached to the cube as shown.
Find the volume and the total surface area of the whole solid (Take π = 3.14).