RD Sharma 2020 solution class 9 chapter 20 Surface Area and Volume of a Right Circular Cone Exercise 20.2

Exercise 20.2

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Question 1:

Find the volume of a right circular cone with:

(i) radius 6 cm, height 7 cm.

(ii) radius 3.5 cm, height 12 cm

(iii) height 21 cm and slant height 28 cm.

Answer 1:

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

Volume =

(i) Substituting the values of r = 6 cm and h = 7 cm in the above equation and using

Volume =

= (22) (2) (6)

= 264

Hence the volume of the given cone with the specified dimensions is

(ii) Substituting the values of r = 3.5 cm and h =12 cm in the above equation and using

Volume =

= (22) (0.5) (3.5) (4)

= 154

Hence the volume of the given cone with the specified dimensions is

(iii) In a cone, the vertical height ‘h’ is given as 21 cm and the slant height ‘l’ is given as 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 above equation and using

Volume =

= (22) (343)

= 7546

Hence the volume of the given cone with the specified dimensions is

Question 2:

Find the capacity in litres of a conical vessel with:

(i) radius 7 cm, slant height 25 cm

(ii) height 12 cm, slant height 13 cm.
 

Answer 2:

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

Volume =

(i) In a cone, the base radius ‘r’ is given as 7 cm and the slant height ‘l’ is given as 25 cm.

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

We know that in a cone

=

=

=

= 24

Therefore the vertical height is, h = 24 cm.

Substituting the values of r = 7 cm and h = 24 cm in the above equation and using

Volume =

= (22) (7) (8)

= 1232

Hence the volume of the given cone with the specified dimensions is

(ii) In a cone, the vertical height ‘h’ is given as 12 cm and the slant height ‘l’ is given as 13 cm.

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

We know that in a cone

=

=

=

= 5

Therefore the base radius is, r = 5 cm.

Substituting the values of r = 5 cm and h = 12 cm in the above equation and using

Volume =

= 314.28

Hence the volume of the given cone with the specified dimensions is

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Question 3:

Two cones have their heights in the ratio 1 : 3 and the radii of their bases in the ratio 3 : 1. Find the ratio of their volumes.

Answer 3:

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

Volume =

Let the base radius and the height of the two cones be and respectively.

It is given that the ratio between the heights of the two cones is 1: 3.

Since only the ratio is given, to use them in our equation we introduce a constant ‘k’.

So, = 1k

= 3k

It is also given that the ratio between the base radiuses of the two cones is 3: 1.

Again, since only the ratio is given, to use them in our equation we introduce another constant ‘p’.

So, = 3p

= 1p

Substituting these values in the formula for volume of cone we get,

=

=

Hence we see that the ratio between the volumes of the two given cones is

Question 4:

The radius and the height of a right circular cone are in the ratio 5 : 12. If its volume is 314 cubic metre, find the slant height and the radius (Use $\mathrm{\pi } = 3.14$).

Answer 4:

It is given that the ratio between the radius ‘r’ and the height ‘h’ of the cone is 5: 12.

Since only the ratio is given, to use them in an equation we introduce a constant ‘k’.

So,

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

Volume =

The volume of the cone is given as

Substituting the values of and and using in the formula for the volume of a cone,

Volume =

314 =

= 1

k = 1

Therefore the actual value of the base radius is r = 5 m and h = 12 m.

Hence the radius of the cone is

We are given that r = 5 m and h = 12 m. We find l using the relation

=

=

=

= 13.

Therefore, the slant height of the given cone is

Hence the radius of cone and slant height is 5 m and 13 m respectively

Question 5:

The radius and height of a right circular cone are in the ratio 5 : 12 and its volume is 2512 cubic cm. Find the slant height and radius of the cone.
(Use $\mathrm{\pi } = 3.14$).

Answer 5:

It is given that the ratio between the radius ‘r’ and the height ‘h’ of the cone is 5: 12.

Since only the ratio is given, to use them in an equation we introduce a constant ‘k’.

So,

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

Volume =

The volume of the cone is given as

Substituting the values of and and using in the formula for the volume of a cone,

Volume =

2512 =

= 8

k = 2

Therefore the actual value of the base radius is r = 10 cm and h = 24 cm.

Hence the radius of the cone is

We are given that r = 10 cm and h = 24 cm. We find l using the relation

=

=

=

= 26

Therefore the slant height of the given cone is

Hence the radius and slant height of the cone are 10 cm and 26 cm respectively

Question 6:

The ratio of volumes of two cones is 4 : 5 and the ratio of the radii of their bases is 2 : 3, Find the ratio of their vertical heights.

Answer 6:

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

Volume =

Let the volume, base radius and the height of the two cones be and respectively.

It is given that the ratio between the volumes of the two cones is 4: 5.

Since only the ratio is given, to use them in our equation we introduce a constant ‘k’.

So, = 4k

= 5k

It is also given that the ratio between the base radiuses of the two cones is 2: 3.

Again, since only the ratio is given, to use them in our equation we introduce another constant ‘p’.

So, = 2p

= 3p

Substituting these values in the formula for volume of cone we get,

=

=

=

=

Therefore the ratio between the heights of the two cones is

Question 7:

A cylinder and a cone have equal radii of their bases and equal heights. Show that their volumes are in the ratio 3 : 1.

Answer 7:

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

Volume of cone =

And, the formula of the volume of a cylinder with base radius ‘r’ and vertical height ‘h’ is given as

Volume of cylinder =

Now, substituting these to arrive at the ratio between the volume of a cylinder and the volume of a cone, we get

= $\frac{{\mathrm{\pi r}}^2\mathrm{h}}{{\displaystyle \frac{1}{3}}{\mathrm{\pi r}}^2\mathrm{h}}$

=

Hence it is shown that the ratio between the volumes of a cylinder and a cone with the same base radius and the same height is indeed

Question 8:

If the radius of the base of a cone is halved, keeping the height same, what is the ratio of the volume of the reduced cone to that of the original cone?

Answer 8:

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

Volume of cone =

Now, let another cone have the same height, that is ‘h’, but the base radius of this cone is half that of the previous one we have talked about, that is ‘’

Now,

The volume of this new cone =

=

Now the ratio between the old cone and the new one would be,

= $\frac{{\displaystyle \frac{1}{3}}{\mathrm{\pi r}}^2\mathrm{h}}{{\displaystyle \frac{\pi r^2\mathrm{h}}{12}}}$=

=

=

Hence the ratio between the volumes of the modified cone and the original cone is

Question 9:

A heap of wheat is in the form of a cone of diameter 9 m and height 3.5 m. Find its volume. How much canvas cloth is required to just cover the heap? (Use $\mathrm{\pi } = 3.14$).

Answer 9:

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

Volume of cone =

Here, the diameter is given as 9 m. From this we get the base radius as r = 4.5 m.

Substituting the values of r = 4.5 m and h = 3.5 m in the above equation and using π = 3.14

Volume =

=

= 74.1825

Hence the volume of the given cone with the specified dimensions is

The amount of canvas required to cover the conical heap would be equal to the curved surface area of the conical heap.

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

To find the slant height ‘l’ to be used in the formula for Curved Surface Area we use the following relation

Slant height, l =

=

=

=

Hence the slant height l of the conical heap is m.

Now, substituting the values of r = 4.5 m and slant height l = m and using in the formula of C.S.A,

We get Curved Surface Area =

= 80.55

Hence the amount of canvas required to just cover the heap would be

Question 10:

Find the weight of a solid cone whose base is of diameter 14 cm and vertical height 51 cm, supposing the material of which it is made weighs 10 grams per cubic cm.

Answer 10:

To find the weight of the cone we first need to find its volume.

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

Volume of cone =

Here, the diameter is given as 14 cm. From this we get the base radius as r = 7 m.

Substituting the values of r = 7 cm and h = 51 cm in the above equation and using

Volume =

= (22) (7) (17)

= 2618

Hence the volume of the given cone with the specified dimensions is 2618 m³

Now, it is given that material of which the cone is made up of weighs 10 grams per cubic meter.

Hence the entire weight of the cone = (Volume of the cone) (10)

= (2618) (10)

= 26180 gram

Hence the weight of the cone is

Question 11:

A right angled triangle of which the sides containing the right angle are 6.3 cm and 10 cm in length, is made to turn round on the longer side. Find the volume of the solid, thus generated. Also, find its curved surface area.

Answer 11:

When you rotate a right triangle about one of its sides containing the right angle the solid so formed will be a cone.

Here the right triangle has sides 6.3 cm and 10 cm and it is said that the right triangle is rotated about its longer side. So here it will be the side of 10 cm length.

So, the height of the cone thus formed will be ‘h’ = 10 cm, and the radius ‘r’ = 6.3 cm.

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

Volume of cone =

Substituting the values of r = 6.3 cm and h = 10 cm in the above equation and using

Volume =

=

= 415.8

Hence the volume of the given cone with the specified dimensions is

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

To find the slant height ‘l’ to be used in the formula for Curved Surface Area we use the following relation

Slant height, l =

=

=

=

Hence the slant height l of the cone is cm.

Now, substituting the values of r = 6.3 cm and slant height l = cm and using in the formula of C.S.A,

We get Curved Surface Area =

= 233.8

Hence the curved surface area of the so formed cone is

Question 12:

Find the volume of the largest right circular cone that can be fitted in a cube whose edge is 14 cm.

Answer 12:

The largest cone that can be fitted into a cube will have its height and base diameter equal to the edge of the cube.

Here the edge of the cube is given as 14 cm.

So the dimensions of the cube with the maximum area would be ‘h’ = 14 cm and base radius ‘r’ = 7 cm.

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

Volume of cone=

Substituting the values of r = 7 cm and h = 14 cm in the above equation and using

Volume =

=

= 718.66

Hence the volume of the largest cone that can be fit into a cube with edge 14 cm is

Question 13:

The volume of a right circular cone is 9856 cm³. If the diameter of the base is 28 cm, find:

(i) height of the cone

(ii) slant height of the cone

(iii) curved surface area of the cone.
 

Answer 13:

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 diameter of the base is 28 cm. Hence the base radius ‘r’ = 14 cm. The volume of the cone is also given as 9856 cm³

We can now find the height of the cone by using the formula for the volume of a cone.

h =

=

= 48

Hence the height of the given cone is

To find the slant height ‘l’ to be used in the formula for Curved Surface Area we use the following relation

Slant height, l =

=

=

=

= 50

Hence the slant height l of the cone is

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

Curved Surface Area =

Now, substituting the values of r = 14 cm and slant height l = 50 cm and using in the formula of C.S.A,

Curved Surface Area =

= 2200

Hence the curved surface area of the given cone is

Question 14:

A conical pit of top diameter  3.5 m is 12 m deep. What is its capacity in kilolitres?
 

Answer 14:

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 top diameter is 3.5 m. Hence the radius of the conical pit is m.

Substituting the values of r = m and h = 12 m in the above equation and using we get

Hence the volume of the conical pit is 38.5 m³ or

Question 15:

Monica has a piece of Canvas whose area is 551 m². She uses it to have a conical tent made, with a base radius of 7 m. Assuming that all the stitching margins and wastage incurred while cutting, amounts to approximately 1 m². Find the volume of the tent that can be made with it.

Answer 15:

Given that out of the 551 m², 1 m² has to be used for stitching, etc we are left with 550 m² of canvas to make a tent.

The amount of canvas needed to make the conical tent would be equal to the curved surface area of the conical tent.

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

Here the C.S.A = 550 m² and the base radius ‘r’ = 7 m. We can get the slant height ‘l’ of the tent by using the formula for curved surface area.

l =

=

= 25

Hence the slant height of the conical tent is 25 m.

The height ‘h’ can be found out using the relation between r, l and h.

We know that in a cone

=

=

=

= 24

Hence the height of the conical tent is 24 m.

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

Volume of cone =

Substituting the values of r = 7 m and h = 24 m in the above equation and using we get,

Volume =

= (22) (7) (8)

= 1232

Hence the volume of the conical tent that can be made out of the given canvas with the given dimensions is