Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.)
Solution:
A figure is drawn below according to the given data to visualize the model.
From the figure, it can be seen that the volume of the model includes the volume of the cylindrical part and both the identical conical parts.
The volume of a solid is the space occupied inside the solid or the capacity a solid can hold.
Volume of the model = Volume of the 2 conical parts + Volume of the cylindrical part
As the length of the model includes the height of the cylindrical part and the height of both the identical conical parts,
Length of the model = Height of the cylindrical part + 2 × Height of the conical part
We will find the volume of the model by using formulae;
Volume of the cylinder = πr2h₁, where r and h₁ are the radius and height of the cylinder respectively.
Volume of the cone = 1/3 πr2h₂, where r and h₂ are the radius and height of the cone respectively.
Height of each conical part, h₂ = 2 cm
Height of cylindrical part = Length of the model - 2 × Height of the conical part
h₁ = 12 cm - 2 × 2 cm = 8 cm
Diameter of the model, d = 3 cm
Radius of cylindrical part = radius of conical part = r = 3/2 cm = 1.5 cm
Volume of the model = 2 × Volume of the conical part + Volume of the cylindrical part
= 2 × 1/3 πr2h₂ + πr2h₁
= πr2 (2/3 h₂ + h₁)
= 22/7 × 1.5 cm × 1.5 cm × (2/3 × 2 cm + 8 cm)
= 22/7 × 1.5 cm × 1.5 cm × 28/3 cm
= 66 cm3
Thus, the volume of air in the model is 66 cm3.
☛ Check: Class 10 Maths NCERT Solutions Chapter 13
Video Solution:
Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.)
NCERT Solutions Class 10 Maths Chapter 13 Exercise 13.2 Question 2
Summary:
Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, the volume of air contained in the model that Rachel made was 66 cm3.
☛ Related Questions:
- A gulab jamun, contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm (see Fig. 13.15).
- 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 depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand (see Fig. 13.16).
- A vessel is in the form of an 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 brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.
- A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm3of iron has approximately 8 g mass. (Use π = 3.14)