# 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

Thus, we will work on the assumption that the outer and inner dimensions of the model are nearly the same.

Therefore, Volume of air in the model = volume of the model based on outer dimensions

As the length of the model includes the height of the cylindrical part and the height of both the identical conical parts and also the diameter of the cylindrical part and conical parts are the same as the diameter of the model.

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 = πr^{2}h_{1}, where r and h_{1} are the radius and height of the cylinder respectively.

Volume of the cone = 1/3 πr^{2}h_{2}, where r and h_{2} are the radius and height of the cone respectively.

Height of each conical part, h_{2} = 2 cm

Height of cylindrical part = Length of the model - 2 × Height of the conical part

h_{1} = 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 πr^{2}h_{2}+ πr^{2}h_{1}

= πr^{2} (2/3 h_{2}+ h_{1})

= 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 cm^{3}

Thus, the volume of air in the model is 66 cm^{3}.

**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

### NCERT Solutions Class 10 Maths - Chapter 13 Exercise 13.2 Question 2 :

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

The volume of air contained in Rachel's cylindrical shaped model with two cones attached at its two ends is 66 cm^{3}