In this blog, we will discuss the frustum of a cone. first of all we will see what is a frustum.
Let us take some clay and make a right circular cone. Cut it with a knife parallel to its base. Remove the smaller cone. What are we left with? We are left with a solid called a frustum of the cone.
How can we find the surface area and volume of a frustum of a cone? For this, we will go with this example.
1. The radii of the ends of a frustum of a cone 45 cm high are 28 cm and 7 cm. Find its volume, the curved surface area, and the total surface area
Solution: The frustum can be viewed as a difference of two right circular cones OAB and OCD. Let the height (in cm) of the cone OAB be h1 and its slant height l1,i.e., OP = h1 and OA = OB = l1. Let h2 be the height of cone OCD and l2 its slant height. We have: r1= 28 cm, r2= 7 cm and the height of frustum (h) = 45 cm. Also, h1 = 45 + h2 call this one. We first need to determine the respective heights h1 and h2 of the cone OAB and OCD. Since the triangles OPB and OQD are similar, we have h1 upon h2 = 28/7 = 4/1. Call this second. From first and second, we get h2 = 15 and h1 = 60.
Now, the volume of the frustum = volume of the cone OAB – volume of the cone OCD
Let h be the height, l the slant height and r1 and r2 the radii of the ends (r1 > r2) of the frustum of a cone. Then we can directly find the volume, the curved surface area and the total surface area of frustum by using these formulae.
Let us apply these formulae in some examples.
1. Hanumappa and his wife Gangamma are busy making jaggery out of sugarcane juice. They have processed the sugarcane juice to make the molasses, which is poured into moulds in the shape of a frustum of a cone having the diameters of its two circular faces as 30 cm and 35 cm and the vertical height of the mould is 14 cm. If each cm3 of molasses has mass of about 1.2 g, find the mass of the molasses that can be poured into each mould.
2. An open metal bucket is in the shape of a frustum of a cone, mounted on a hollow cylindrical base made of the same metallic sheet. The diameters of the two circular ends of the bucket are 45 cm and 25 cm, the total vertical height of the bucket is 40 cm and that of the cylindrical base is 6cm. Find the area of the metallic sheet used to make the bucket, where we do not take into account the handle of the bucket. Also, find the volume of water the bucket can hold.