RD Sharma 2020 solution class 9 chapter 20 Surface Area and Volume of a Right Circular Cone VSAQS

VSAQS

Page-20.26

Question 1:

The height of a cone is 15 cm. If its volume is 500$\mathrm{\pi }$ cm³, then find the radius of its base.

Answer 1:

The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as

Volume of cone =

It is given that the height of the cone is ‘h’ = 15 cm and that the volume of the cone is

We can now find the radius of base ‘r’ by using the formula for the volume of a cone.

=

=

= 100

= 10

Hence the radius of the base of the given cone is

Question 2:

If the volume of a right circular cone of height 9 cm is 48$\mathrm{\pi }$ cm³, find the diameter of its base.

Answer 2:

The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as

Volume of cone =

It is given that the height of the cone is ‘h’ = 9 cm and that the volume of the cone is 48π cm³

We can now find the radius of base ‘r’ by using the formula for the volume of a cone.

=

=

= 16

= 4

Hence the radius of the base of the cone with given dimensions is ‘r’ = 4 cm.

The diameter of base is twice the radius of the base.

Hence the diameter of the base of the cone is

Question 3:

If the height and slant height of a cone are 21 cm and 28 cm respectively. Find its volume.

Answer 3:

The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as

Volume of cone=

The vertical height is given as ‘h’ = 21 cm, and the slant height is given as ‘l’ = 28 cm.

To find the base radius ‘r’ we use the relation between r, l and h.

We know that in a cone 

=

=

=

Therefore the base radius is, r = cm.

Substituting the values of r = cm and h = 21 cm in the formula for volume of a cone.

Volume = $\frac{{\mathrm{\pi r}}^{{}_2}\mathrm{h}}{3}=\frac{\mathrm{\pi }\times {\left(\sqrt{343}\right)}^2\times 21}{3}$
=2401$\mathrm{\pi }$

Hence the volume of the given cone with the specified dimensions is $2401\mathrm{\pi } {\mathrm{cm}}^3$.

Question 4:

The height  of a conical vessel is 3.5 cm. If its capacity is 3.3 litres of milk. Find its diameter of its base.

Answer 4:

The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as

Volume of cone =

It is given that the height of the cone is ‘h’ = 3.5 cm and that the volume of the cone is 3.3 liters

We know that,

1 liter = 1000 cubic centimeter

Hence, the volume of the cone in cubic centimeter is. 

We can now find the radius of base ‘r’ by using the formula for the volume of a cone, while using

=

=

= 900

= 30

Hence the radius of the base of the cone with given dimensions is ‘r’ = 30 cm.

The diameter of base is twice the radius of the base.

Hence the diameter of the base of the cone is

Question 5:

If the radius and slant height of a cone are in the ratio 7 : 13 and its curved surface area is 286 cm², find its radius.

Answer 5:

It is given that the curved surface area (C.S.A) of the cone is 286 cm² and that the ratio between the base radius and the slant height is 7: 13. The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as

Curved Surface Area = πrl

Since only the ratio between the base radius and the slant height is given, we shall use them by introducing a constant ‘k’

So, r = 7k

l = 13k

Substituting the values of C.S.A, base radius, slant height and using in the above equation,

Curved Surface Area, 286 =

286 = 286 k²

1 = k²

Hence the value of k = 1

From this we can find the value of base radius,

r = 7k

r = 7

Therefore the base radius of the cone is

Question 6:

Find the area of canvas required for a conical tent of height 24 m and base radius 7 m.

Answer 6:

The amount of canvas required to make a cone would be equal to the curved surface area of the cone.

The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as 

Curved Surface Area =

It is given that the vertical height ‘h’ = 24 m and base radius ‘r’ = 7 m.

To find the slant height ‘l’ we use the following relation

Slant height, l =

=

=

=

l = 25

Hence the slant height of the given cone is 25 m.

Now, substituting the values of r = 7 m and slant height l = 25 m and using in the formula of C.S.A,

We get 

Curved Surface Area =

= (22) (25)

= 550 

Therefore the Curved Surface Area of the cone is

Question 7:

Find the area of metal sheet required in making a closed hollow cone of base radius 7 cm and height 24 cm.

Answer 7:

The area of metal sheet required to make this hollow closed cone would be equal to the total surface area of the cone.

The formula of the total surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as 

Total Surface Area =

It is given that the vertical height ‘h’ = 24 cm and base radius ‘r’ = 7 cm.

To find the slant height ‘l’ we use the following relation

Slant height, l =

=

=

=

l = 25

Hence the slant height of the given cone is 25 cm.

Now, substituting the values of r = 7 cm and slant height l = 25 cm and using in the specified formula,

Total Surface Area =

= (22) (32)

= 704

Therefore the total area of the metal sheet required to make the closed hollow cone is equal to

Question 8:

Find the length of cloth used in making a conical pandal of height 100 m and base radius 240 m, if the cloth is 100$\mathrm{\pi }$ m wide.

Answer 8:

The area of cloth required to make the conical pandal would be equal to the curved surface area of the cone.

The formula of the curved surface area of a cone with base radius ‘r’ and slant height ‘l’ is given as 

Curved Surface Area = πrl

It is given that the vertical height ‘h’ = 100 m and base radius ‘r’ = 240 m.

To find the slant height ‘l’ we use the following relation

Slant height, l =

=

=  

=

l = 260

Hence the slant height of the given cone is 260 m.

Now, substituting the values of r = 240 m and slant height l = 260 m in the formula for C.S.A,

We get 

Curved Surface Area =

=

Hence the area of the cloth required to make the conical pandal would be m²

It is given that the cloth is 100π wide. Now, we can find the length of the cloth required by using the formula,

Length of the canvas required =

=

= 624

Hence the length of the cloth that is required is