3.Surface Area & Volumes – Frustum of a Cone Notes NCERT Solutions for CBSE Class 10 Mathmatics : EduMple

Books Name
Mathematics Book

Publication
ACERISE INDIA

Course
CBSE Class 10

Subject
Mathmatics

Surface Areas of Frustum of Cone

Look at the following figure.

In day to day life, we come across various shapes similar to this, such as a glass, a bucket, a funnel, etc. This shape is known as frustum of cone.

Now, can we describe what a frustum is?

The solid obtained by cutting a cone by a plane parallel to its base is known as frustum of a cone.

We can observe from the following figure that a cone is sliced by a plane parallel to the base. After removing the smaller cone, we obtain a solid which is known as frustum of cone.

The given figure shows an iron tub and its dimensions. Its exterior curved surface is to be polished. Can we find the area of iron tub to be polished?

To find the total surface area of a frustum, we just need to add the areas of its two circular faces to its curved surfaces area.

Therefore,

Total surface area of frustum = Curved surface area + Area of two circular faces Total surface area of a frustum = π(r1 + r2)l + πr₁² + πr₂²

Let us look at some examples to understand the concept better.

Example 1: A shuttle cock, which is used for playing badminton, has the shape of a frustum mounted on a hemisphere as shown in the following figure. The dimensions are also shown in the figure. Find the outer curved surface area of the shuttle cock. (Use π = 3.14)

Solution:

It is given that,

r₁ = radius of upper end of frustum = 3.5 cm

r₂= radius of lower end of frustum = 1.5 cm (Also the radius of the hemisphere)

h = height of the frustum = cm

Now, slant height,

= 6.32 cm

Outer curved surface area of shuttle cock = C.S.A. of frustum + C.S.A. of hemisphere

= +

= 99.22 + 14.13

= 113.35 cm²

Thus, the required surface area is 113.35 cm².

Example 2: A solid is in the shape of the frustum of a cone. The radii of its circular ends are 7 cm and 14 cm. The total surface area of the solid is 2420 cm2. What is the height of the solid?

Solution:

The solid is in the shape of a conical frustum with radii 7 cm and 14 cm and total surface area 2420 cm2.

Let r1 and r2 be the radii of the solid. Let l and h be its slant height and height respectively.

The total surface area of the frustum is given by .

Here, r₁ = 14 cm and r₂= 7 cm

Now,

Thus, the height of the solid is 24 cm.

Example 3: The given figure shows a wooden shape which is in the shape of a frustum of a cone. The diameters of the upper and lower circular part are 4 ft and 8 ft respectively. Its curved surfaces are painted with red colour. If the cost of painting is Rs 660 at the rate of Rs 5 per ft², then find the height of the wooden shape.

Solution:

The cost of painting is given to be Rs 660.

The area that was painted is the curved surface area of the wooden shape.

Also, it is given that the rate of painting is Rs 5 per ft2.

∴ C.S.A. of the wooden shape = = 132 ft²

We know, C.S.A. = π (r1 + r2) l, where r1 and r2 are the radii of the ends of the frustum shaped wood and l is the slant height.

We have, π (r₁ + r₂) l = 132 ft²

Let h be the height of the wooden shape.

Then, we have

Thus, the height of the wooden shape is ft.

Example 4: Adam brought a conical pancake (a cake eaten in USA) that was 16 cm high with vertical angle of 60° and kept it in a refrigerator. After sometime, his cousin Susan came and cut the pancake into two parts with the help of a knife at the middle point of its height and took the smaller part. She also covered the tissue paper entirely around the remaining cake. Calculate the least amount of paper in terms of π that she required to cover the pancake.

Solution:

Let us consider the following figure in order to understand the question properly.

Adam brought the conical pancake OAB whose vertical angle ∠AOB = 60°.

Here, height of the pancake, OP = 16 cm

Let us consider Q as the mid-point of OP. Susan cut the pan cake into two parts through Q.

Therefore, we have OQ = cm

Also, ∠AOP and COQ =

After cutting conical pancake OCD from pancake OAB, the remaining pancake is in the shape of a frustum.

In ΔAOP,

⇒ cm

Also in Δ COQ,

⇒ cm

Height of the frustum, h = PQ = OP − OQ = 16 − 8 = 8 cm

Now, slant height of frustum

= cm

T.S.A. of the frustum = +

= cm²

Thus, Susan required cm²of paper to cover the remaining pancake.

Volume Of A Frustum Of A Cone

Suppose there is a container made up of metal sheet in the form of a frustum of a cone of height 14 cm with diameter of its lower and upper ends as 50 cm and 20 cm respectively.

Can we find the cost of honey which can completely fill the container at the rate of Rs 150 per litre?

Let us look at some more examples to understand the concept better.

Example 1: A metallic right circular cone, 14 cm high and whose vertical angle is 60°, is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained is drawn into a wire of diameter mm, then find the length of the wire.

Solution:

Let us consider the following figure in order to understand the question properly.

Let OAB be the cone whose vertical angle ∠AOB = 60°. Let r₁ and r₂ be the radii of the lower and upper end of the frustum.

Here, height of the cone, OP = 14 cm

Let us consider Q as the mid-point of OP.

After cutting the cone into two parts through Q,

OQ =

Also, ∠AOP and ∠COQ =

After cutting cone OCD from cone OAB, the remaining solid obtained is a frustum.

In ΔAOP,

⇒ cm

Also in ΔCOQ,

⇒ cm

Height of the frustum, h = PQ = OP − OQ = 14 − 7 = 7 cm

For the cylinder shaped wire,

Radius, r = = mm =cm

Let the length of the wire be H.

It is given that the frustum is drawn into a cylindrical wire.

∴ Volume of Cylinder = Volume of Frustum

Thus, the length of the wire drawn from the frustum is 784 m.

Example 2: A hollow cone of height 30 cm is cut by a plane parallel to the base, and the upper portion is removed. If the volume of the remainder (frustum of cone) is of the volume of the whole cone, then find the ratio of the altitudes of the frustum and the cone that is removed.

Solution:

Let us consider that cone OCD ( radius r₂, height h₂, and slant height l₂) is removed from a cone OAB of radius r₁, height h₁, and slant height l₁, where circular base CD is parallel to circular base AB.