Rd Sharma 2020 2021 Solutions for Class 9 Maths Chapter 20 Surface Area And Volume Of A Right Circular Cone are provided here with simple step-by-step explanations. These solutions for Surface Area And Volume Of A Right Circular Cone are extremely popular among Class 9 students for Maths Surface Area And Volume Of A Right Circular Cone Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma 2020 2021 Book of Class 9 Maths Chapter 20 are provided here for you for free. You will also love the ad-free experience on Meritnationâ€™s Rd Sharma 2020 2021 Solutions. All Rd Sharma 2020 2021 Solutions for class Class 9 Maths are prepared by experts and are 100% accurate.

#### Page No 20.20:

#### 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:

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

#### Page No 20.20:

#### 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:

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

#### Page No 20.21:

#### 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:

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, = 1*k*

= 3*k*

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, = 3*p*

= 1*p*

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

#### Page No 20.21:

#### 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:

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

#### Page No 20.21:

#### 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:

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

#### Page No 20.21:

#### 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:

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, = 4*k*

= 5*k*

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, = 2*p*

= 3*p*

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

=

=

=

=

Therefore the ratio between the heights of the two cones is

#### Page No 20.21:

#### 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:

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

#### Page No 20.21:

#### 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:

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

#### Page No 20.21:

#### 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:

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

#### Page No 20.21:

#### 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:

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 m^{3}

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

#### Page No 20.21:

#### 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:

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

#### Page No 20.21:

#### Question 12:

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

#### Answer:

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

#### Page No 20.21:

#### Question 13:

The volume of a right circular cone is 9856 cm^{3}. 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:

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 cm^{3}

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

#### Page No 20.21:

#### Question 14:

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

#### Answer:

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 m^{3} or

#### Page No 20.21:

#### Question 15:

Monica has a piece of Canvas whose area is 551 m^{2}. 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 m^{2}. Find the volume of the tent that can be made with it.

#### Answer:

Given that out of the 551 m^{2}, 1 m^{2} has to be used for stitching, etc we are left with 550 m^{2} 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.

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area = *πrl*

Here the C.S.A = 550 m^{2} 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

#### Page No 20.23:

#### Question 1:

*Mark the correct alternative in each of the following:*

The number of surfaces of a cone has, is

(a) 1

(b) 2

(c) 3

(d) 4

#### Answer:

The surfaces or faces that a cone has are :

(1) Base

(2) Slanted Surface

So, the number of surfaces that a cone has is 2.

Hence the correct choice is (b).

#### Page No 20.23:

#### Question 2:

The area of the curved surface of a cone of radius 2r and slant height $\frac{l}{2}$, is

(a) $\mathrm{\pi}rl$

(b) 2$\mathrm{\pi}rl$

(c) $\frac{1}{2}\mathrm{\pi}rl$

(d) $\mathrm{\pi}(r+l)r$

#### Answer:

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 base radius is given as ‘2*r*’ and the slant height is given as ‘’

Substituting these values in the above equation we have

Curved Surface Area =

= *πrl*

Hence the correct choice is (a).

#### Page No 20.24:

#### Question 3:

The total surface area of a cone of radius $\frac{r}{2}$ and length 2*l*, is

(a) 2$\pi r(\mathit{1}+r)$

(b) $\mathrm{\pi r}\left(1+\frac{\mathrm{r}}{4}\right)$

(c) $\mathrm{\pi}r(1+r)$

(d) 2$\mathrm{\pi}r\mathrm{l}$

#### Answer:

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

Total Surface Area =

Here it is given that the base radius is ‘’ and that the slant height is ‘2*l*’.

Substituting these values in the above equation we have

Total Surface Area =

=

Hence the correct choice is (b).

#### Page No 20.24:

#### Question 4:

A solid cylinder is melted and cast into a cone of same radius. The heights of the cone and cylinder are in the ratio

(a) 9 : 1

(b) 1 : 9

(c) 3 : 1

(d) 1 : 3

#### Answer:

Since the cylinder is re cast into a cone both their volumes should be equal.

So, let Volume of the cylinder = Volume of the cone

= *V*

It is also given that their base radii are the same.

So, let Radius of the cylinder = Radius of the cone

= *r*

Let the height of the cylinder and the cone be and respectively.

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

Volume of cone =

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

Volume of cylinder =

So we have

$\frac{Volumeofcone}{Volumeofcylinder}=\frac{{\displaystyle \frac{1}{3}}{\mathrm{\pi r}}^{2}{\mathrm{h}}_{\mathrm{cone}}}{\pi {r}^{2}{\mathrm{h}}_{cylinder}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{V}{V}=\frac{{\displaystyle \frac{1}{3}{\mathrm{h}}_{cone}}}{{\mathrm{h}}_{cylinder}}\phantom{\rule{0ex}{0ex}}\Rightarrow 1=\frac{{\displaystyle \frac{1}{3}{\mathrm{h}}_{cone}}}{{\mathrm{h}}_{cylinder}}$

Hence the correct choice is option (c).

#### Page No 20.24:

#### Question 5:

If the radius of the base of a right circular cone is 3r and its height is equal to the radius of the base, then its volume is

(a) $\frac{1}{3}{\mathrm{\pi r}}^{3}$

(b) $\frac{2}{3}{\mathrm{\pi r}}^{3}$

(c) $3{\mathrm{\pi r}}^{3}$

(d) 9${\mathrm{\pi r}}^{3}$

#### Answer:

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

Volume of cone =

Here it is given that the base radius is ’3*r*’ and that the vertical height is ‘3*r*’

Substituting these values in the above equation we get

Volume of cone =

=

Hence the correct answer is option (d).

#### Page No 20.24:

#### Question 6:

If the volume of two cones are in the ratio 1 : 4 and their diameters are in the ratio 4 : 5, then the ratio of their heights, is

(a) 1 : 5

(b) 5 : 4

(c) 5 : 16

(d) 25 : 64

#### Answer:

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 1 : 4.

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

So, = 1*k*

= 4*k*

It is also given that the ratio between the base diameters of the two cones is 4 : 5.

Hence the ratio between the base radius will also be 4 : 5.

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

So, = 4*p*

= 5*p*

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

=

=

=

=

Hence the correct answer is option (d).

#### Page No 20.24:

#### Question 7:

The curved surface area of one cone is twice that of the other while the slant height of the latter is twice that of the former. The ratio of their radii is

(a) 2 : 1

(b) 4 : 1

(c) 8 : 1

(d) 1 : 1

#### Answer:

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area =

Now there are two cones with base radius, slant height and Curved Surface Area (C.S.A) as , , & , , respectively.

It is given that = 2() and also that = 2(). Or this can also be written as

=

=

=

=

=

Hence the correct choice is option (b).

#### Page No 20.24:

#### Question 8:

If the height and radius of a cone of volume V are doubled, then the volume of the cone, is

(a) 3 *V*

(b) 4 *V*

(c) 6 *V*

(d) 8 *V*

#### Answer:

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

Volume of cone = = *V*

Since it is given that the radius and height are doubled we have the radius as ‘2*r*’ and the vertical height as ‘2*h*’

So now,

Volume of modified cone =

=

= 8*V*

Hence the correct answer is option (d).

#### Page No 20.24:

#### Question 9:

The ratio of the volume of a right circular cylinder and a right circular cone of the same base and height, is

(a) 1 : 3

(b) 3 : 1

(c) 4 : 3

(d) 3 : 4

#### Answer:

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

=

=

Hence the correct answer is option (b).

#### Page No 20.24:

#### Question 10:

A right circular cylinder and a right circular cone have the same radius and the same volume. The ratio of the height of the cylinder to that of the cone is

(a) 3 : 5

(b) 2 : 5

(c) 3 : 1

(d) 1 : 3

#### Answer:

It is given that the volumes of both the cylinder and the cone are the same.

So, let Volume of the cylinder = Volume of the cone = *V*

It is also given that their base radii are the same.

So, let Radius of the cylinder = Radius of the cone

= *r*

Let the height of the cylinder and the cone be and respectively.

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

Volume of cone =

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

Volume of cylinder =

So we have

Hence the correct choice is option (d).

#### Page No 20.24:

#### Question 11:

The diameters of two cones are equal. If their slant heights are in the ratio 5 : 4, the ratio of their curved surface areas, is

(a) 4 : 5

(b) 25 : 16

(c) 16 : 25

(d) 5 : 4

#### Answer:

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area = *πrl*

Now there are two cones with base radius and slant heights as , & , respectively.

The ratio between slant heights of the two cones is given as 5 : 4, we shall use them by introducing a constant ‘k’

So, now = 5k

= 4k

Since the base diameters of both the cones are equal we get that = =

Using these values we shall evaluate the ratio between the curved surface areas of the two cones

=

=

=

Hence the correct answer is option** **(d).

#### Page No 20.24:

#### Question 12:

If the heights of two cones are in the ratio of 1 : 4 and the radii of their bases are in the ratio 4 : 1, then the ratio of their volumes is

(a) 1 : 2

(b) 2 : 3

(c) 3 : 4

(d) 4 : 1

#### Answer:

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 : 4.

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

So, = 1*k*

= 4*k*

It is also given that the ratio between the base radius of the two cones is 4 : 1.

*p*’.

So, = 4*p*

= 1*p*

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

=

=

Hence the correct choice is option (d).

#### Page No 20.24:

#### Question 13:

The slant height of a cone is increased by 10%. If the radius remains the same, the curved surface area is increased by

(a) 10%

(b) 12.1%

(c) 20%

(d) 21%

#### Answer:

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area = *πrl*

Now, it is said that the slant height has increased by 10%.So the new slant height is ‘1.1*l*’

So, now

New Curved Surface Area = 1.1*πrl*

We see that the percentage increase of the Curved Surface Area is 10%

Hence the correct option is (a).

#### Page No 20.24:

#### Question 14:

The height of a solid cone is 12 cm and the area of the circular base is 64$\mathrm{\pi}$cm^{2}. A plane parallel to the base of the cone cuts through the cone 9 cm above the vertex of the cone, the areas of the base of the new cone so formed is

(a) 9$\mathrm{\pi}$ cm^{2}

(b) 16$\mathrm{\pi}$ cm^{2}

(c) 25$\mathrm{\pi}$ cm^{2}

(d) 36$\mathrm{\pi}$ cm^{2}

#### Answer:

If a cone is cut into two parts by a plane parallel to the base, the portion that contains the base is called the frustum of a cone

Let ‘*r*’ be the top radius

‘*R*’ be the radius of the base

‘*h*’ be the height of the frustum

‘*l*’ be the slant height of the frustum.

‘*H*’ be the height of the complete cone from which the frustum is cut

Then from similar triangles we can write the following relationship

Here it is given that the area of the base is 64*π* cm^{2}.

The area of the base with a base radius of ‘*r*’ is given by the formula

Area of base = *πr*^{2}

Substituting the known values in this equation we get

64 *π* = *πr*^{2}

*r*^{2}^{ }= 64

*r *= 8

Hence the radius of the base of the original cone is 8 cm.

So, now let the plane cut the cone parallel to the base at 9 cm from the vertex.

Based on this we get the values as

*R* = 8

*H* = 12

*H – h* = 9

Substituting these values in the relationship mentioned earlier

Hence the radius of the new conical part that has been formed is 6 cm.

And the area of this base of this conical part would be

Area of the base = *πr*^{2}

= 36*π*

Hence the correct choice is option (d).

#### Page No 20.25:

#### Question 15:

If the base radius and the height of a right circular cone are increased by 20%, then the percentage increase in volume is approximately

(a) 60

(b) 68

(c) 73

(d) 78

#### Answer:

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

Volume of cone =

= *V*

It is given that the base radius and the height are increased by 20%. So now the base radius is ‘1.2*r*’ and the height is ‘1.2*h*’.

So,

The volume of the modified cone =

=

= 1.728* V*

Hence the percentage increase in the volume of the cone is 72.8%, which is approximately equal to 73%.

Hence the correct answer is option (c).

#### Page No 20.25:

#### Question 16:

If h, S and V denote respectively the height, curved surface area and volume of a right circular cone, then $3{\mathrm{\pi Vh}}^{3}-{\mathrm{S}}^{2}{\mathrm{h}}^{2}+9{\mathrm{V}}^{2}\phantom{\rule{0ex}{0ex}}$ is equal to

(a) 8

(b) 0

(c) 4$\mathrm{\pi}$

(d) 32${\mathrm{\pi}}^{2}$

#### Answer:

Here we are asked to find the value for a given specific equation which is in terms of *V*,* h *and *S* representing the volume, vertical height and the Curved Surface Area of a cone.

We know $V=\frac{1}{3}{\mathrm{\pi r}}^{2}\mathrm{h}$ and S = $\mathrm{\pi rl}$.

Also, $l=\sqrt{{r}^{2}+{h}^{2}}$

Now, the given equation is $3{\mathrm{\pi Vh}}^{3}-{\mathrm{S}}^{2}{\mathrm{h}}^{2}+9{\mathrm{V}}^{2}\phantom{\rule{0ex}{0ex}}$

So,

$3{\mathrm{\pi Vh}}^{3}-{\mathrm{S}}^{2}{\mathrm{h}}^{2}+9{\mathrm{V}}^{2}\phantom{\rule{0ex}{0ex}}=3\mathrm{\pi}\left(\frac{1}{3}{\mathrm{\pi r}}^{2}\mathrm{h}\right){\mathrm{h}}^{3}-{\left(\mathrm{\pi rl}\right)}^{2}{\mathrm{h}}^{2}+9{\left(\frac{1}{3}{\mathrm{\pi r}}^{2}\mathrm{h}\right)}^{2}\phantom{\rule{0ex}{0ex}}={\mathrm{\pi}}^{2}{\mathrm{r}}^{2}{\mathrm{h}}^{4}-{\mathrm{\pi}}^{2}{\mathrm{r}}^{2}{\mathrm{l}}^{2}{\mathrm{h}}^{2}+9\left(\frac{1}{9}{\mathrm{\pi}}^{2}{\mathrm{r}}^{4}{\mathrm{h}}^{2}\right)\phantom{\rule{0ex}{0ex}}={\mathrm{\pi}}^{2}{\mathrm{r}}^{2}{\mathrm{h}}^{4}-{\mathrm{\pi}}^{2}{\mathrm{r}}^{2}{\mathrm{h}}^{2}{\left(\sqrt{{\mathrm{r}}^{2}+{\mathrm{h}}^{2}}\right)}^{2}+{\mathrm{\pi}}^{2}{\mathrm{r}}^{4}{\mathrm{h}}^{2}\phantom{\rule{0ex}{0ex}}$

$={\mathrm{\pi}}^{2}{\mathrm{r}}^{2}{\mathrm{h}}^{4}-{\mathrm{\pi}}^{2}{\mathrm{r}}^{2}{\mathrm{h}}^{2}{\left(\sqrt{{\mathrm{r}}^{2}+{\mathrm{h}}^{2}}\right)}^{2}+{\mathrm{\pi}}^{2}{\mathrm{r}}^{4}{\mathrm{h}}^{2}\phantom{\rule{0ex}{0ex}}={\mathrm{\pi}}^{2}{\mathrm{r}}^{2}{\mathrm{h}}^{4}-{\mathrm{\pi}}^{2}{\mathrm{r}}^{2}{\mathrm{h}}^{2}\left({\mathrm{r}}^{2}+{\mathrm{h}}^{2}\right)+{\mathrm{\pi}}^{2}{\mathrm{r}}^{4}{\mathrm{h}}^{2}\phantom{\rule{0ex}{0ex}}={\mathrm{\pi}}^{2}{\mathrm{r}}^{2}{\mathrm{h}}^{4}-{\mathrm{\pi}}^{2}{\mathrm{r}}^{4}{\mathrm{h}}^{2}-{\mathrm{\pi}}^{2}{\mathrm{r}}^{2}{\mathrm{h}}^{4}+{\mathrm{\pi}}^{2}{\mathrm{r}}^{4}{\mathrm{h}}^{2}\phantom{\rule{0ex}{0ex}}=0$

Hence the correct choice is option (b).

#### Page No 20.25:

#### Question 17:

If a cone is cut into two parts by a horizontal plane passing through the mid-point of its axis, the axis, the ratio of the volumes of upper and lower part is

(a) 1 : 2

(b) 2 : 1

(c) 1: 7

(d) 1 : 8

#### Answer:

If a cone is cut into two parts by a plane parallel to the base, the portion that contains the base is called the frustum of a cone

Let ‘*r*’ be the top radius

‘*R*’ be the radius of the base

‘*h*’ be the height of the frustum

‘*l*’ be the slant height of the frustum.

‘*H*’ be the height of the complete cone from which the frustum is cut

Then from similar triangles we can write the following relationship

Here, since the plane passes through the midpoint of the axis of the cone we have

*H* = 2* h*

Substituting this in the earlier relationship we have

The volume of the entire cone with base radius ‘*R*’ and vertical height ‘*H*’ would be

Volume of the uncut cone =

Replacing and *H* = 2* h* in the above equation we get

Volume of the uncut cone =

=

Volume of the smaller cone − the top part after the original cone is cut − with base radius ‘*r*’ and vertical height ‘*h*’ would be

Volume of the top part=

Now, the volume of the frustum − the bottom part after the original cone is cut − would be,

Volume of the bottom part= Volume of the uncut cone − Volume of the top part after the cone is cut

= -

Volume of the bottom part=

Now the ratio between the volumes of the top part and the bottom part after the cone is cut would be,

Hence the correct choice is option (c).

#### Page No 20.25:

#### Question 1:

The ratio of the volume of a right circular cylinder and a right circular cone of the same base and height is __________.

#### Answer:

Let *r* be the radius of base and *h* be the height of both right circular cylinder and right circular cone.

∴ Volume of the cylinder, *V*_{1} = *$\mathrm{\pi}{r}^{2}h$*

Volume of the cone, *V*_{2} = $\frac{1}{3}\mathrm{\pi}{r}^{2}h$

Now,

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

⇒ *V*_{1} : *V*_{2} = 3 : 1

Thus, the ratio of the volume of a right circular cylinder and a right circular cone of the same base and height is 3 : 1.

The ratio of the volume of a right circular cylinder and a right circular cone of the same base and height is _____3 : 1_____.

#### Page No 20.25:

#### Question 2:

From a solid right circular cylinder of height *h* and base radius *r*, a conical cavity of the same height and base is scooped out. Then the ratio of the volume of the cone and the remaining solid is __________.

#### Answer:

It is given that, the radius and height of the solid right circular cylinder is *r* and *h*, respectively.

Now,

Radius of the conical cavity scooped out = Radius of the solid right circular cylinder = *r*

Height of the conical cavity scooped out = Height of the solid right circular cylinder = *h*

∴ Volume of the cone, *V*_{1} = $\frac{1}{3}\mathrm{\pi}{r}^{2}h$

Also,

Volume of the remaining solid, *V*_{2}

= Volume of the cylinder − Volume of the conical cavity scooped out

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

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

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

⇒ Volume of the cone, *V*_{1} : Volume of the remaining solid, *V*_{2} = 1 : 2

Thus, the the ratio of the volume of the cone and the remaining solid is 1 : 2.

From a solid right circular cylinder of height *h* and base radius *r*, a conical cavity of the same height and base is scooped out. Then the ratio of the volume of the cone and the remaining solid is _____1 : 2_____.

#### Page No 20.25:

#### Question 3:

The radii of the bases of a cylinder and a cone are in the ratio 3 : 4. If their height are in the ratio 2 : 3, then their volumes are in the ratio ________.

#### Answer:

Let *r* be the radius and *h* be the height of the cylinder & *R* be the radius and *H* be the height of the cone.

It is given that,

$\frac{r}{R}=\frac{3}{4}$ and $\frac{h}{H}=\frac{2}{3}$ .....(1)

Suppose *V*_{1} be the volume of the cylinder and *V*_{2} be the volume of the cone.

$\therefore \frac{{V}_{1}}{{V}_{2}}$

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

$=3\times {\left(\frac{r}{R}\right)}^{2}\times \frac{h}{H}$

$=3\times {\left(\frac{3}{4}\right)}^{2}\times \frac{2}{3}$ [Using (1)]

$=\frac{9}{8}$

∴ Volume of the cylinder, *V*_{1} : Volume of the cone, *V*_{2} = 9 : 8

Thus, the ratio of volume of cylinder to the volume of cone is 9 : 8.

The radii of the bases of a cylinder and a cone are in the ratio 3 : 4. If their height are in the ratio 2 : 3, then their volumes are in the ratio ____9 : 8____.

#### Page No 20.25:

#### Question 4:

A conical tent is 12 m high and the radius of its base is 9 m. The cost of canvas required to make the tent at the rate of â‚ą14 per m^{2}, is __________.

#### Answer:

Radius of the conical tent, *r* = 9 m

Height of the conical tent, *h* = 12 m

Let *l* be the slant height of the conical tent.

$\therefore l=\sqrt{{r}^{2}+{h}^{2}}=\sqrt{{\left(9\right)}^{2}+{\left(12\right)}^{2}}=\sqrt{225}=15\mathrm{m}$

Now,

Area of the canvas required to make the tent = Curved surface area of the conical tent $=\mathrm{\pi}rl=\frac{22}{7}\times 9\times 15=\frac{2970}{7}$ m^{2}

Rate of the canvas = â‚ą14 per m^{2}

∴ Cost of the canvas required to make the tent

= Area of the canvas required to make the tent × Rate of the canvas

$=\u20b9\frac{2970}{7}\times 14$

= â‚ą5,940

Thus, the cost of canvas required to make the tent is â‚ą5,940.

A conical tent is 12 m high and the radius of its base is 9 m. The cost of canvas required to make the tent at the rate of â‚ą14 per m^{2}, is ____â‚ą5,940____.

#### Page No 20.25:

#### Question 5:

A metal cuboid of dimensions 49 m, 22 m and 14 m is melted and cast into 7 identical cylinders of radius 7 m. These cylinders are again melted and cast into cubes such that the side of each cube is equal to half of the height of each cylinder. The numbers of cubes thus formed is __________.

#### Answer:

Let *h* be the height of each cylinder.

Radius of each cylinder, *r* = 7 m

It is given that, the metal cuboid of given dimensions is melted and re-cast into 7 identical cylinders.

∴ 7 × Volume of each cylinder = Volume of the cuboid

$\Rightarrow 7\times \mathrm{\pi}{r}^{2}h=\mathrm{Length}\times \mathrm{Breadth}\times \mathrm{Height}$

$\Rightarrow 7\times \frac{22}{7}\times {\left(7\mathrm{m}\right)}^{2}\times h=49\mathrm{m}\times 22\mathrm{m}\times 14\mathrm{m}$

$\Rightarrow h=\frac{49\times 22\times 14}{22\times 49}=14\mathrm{m}$

So, the height of each cylinder is 14 m.

It is also given that these cylinders are again melted and re-cast into cubes such that the side of each cube is half of the height of each cylinder.

Let the side of each cube be *a*.

$\therefore a=\frac{h}{2}=\frac{14}{2}=7\mathrm{m}$

Suppose the number of cubes formed be *n*.

∴ *n* × Volume of each cube = Volume of 7 identical cylinder = Volume of the cuboid

$\Rightarrow n=\frac{\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{cuboid}}{\mathrm{Volume}\mathrm{of}\mathrm{each}\mathrm{cube}}$

$\Rightarrow n=\frac{49\mathrm{m}\times 22\mathrm{m}\times 14\mathrm{m}}{{\left(7\mathrm{m}\right)}^{3}}$ [Volume of the cube = (Side)^{3}]

$\Rightarrow n=44$

Thus, the number of cubes formed is 44.

A metal cuboid of dimensions 49 m, 22 m and 14 m is melted and cast into 7 identical cylinders of radius 7 m. These cylinders are again melted and cast into cubes such that the side of each cube is equal to half of the height of each cylinder. The numbers of cubes thus formed is _____44_____.

#### Page No 20.25:

#### Question 6:

A solid cylinder and a solid cone have equal bases and equal heights. If the radius and height be in the ratio 4 : 3, the ratio of the total surface area of the cylinder to that of the cone is __________.

#### Answer:

Let *r* be the radius of the solid cylinder and solid cone & *h* be the height of the solid cylinder and solid cone.

It is given that, *r* : *h* = 4 : 3

Suppose *r* = 4*x* units and *h* = 3*x *units, where *x* is constant

Let *l* be the slant height of the cone.

$\therefore l=\sqrt{{r}^{2}+{h}^{2}}=\sqrt{{\left(4x\right)}^{2}+{\left(3x\right)}^{2}}=\sqrt{25{x}^{2}}=5x$ units

Now,

Total surface area of the cylinder = 2*$\mathrm{\pi}$r(r *+ *h*) = 2$\mathrm{\pi}$ × 4*x* × (4*x* + 3*x*) = 56$\mathrm{\pi}$*x*^{2} sq. units

Total surface area of the cone = *$\mathrm{\pi}$r(r *+ *l*) = $\mathrm{\pi}$ × 4*x* × (4*x* + 5*x*) = 36$\mathrm{\pi}$*x*^{2} sq. units

$\therefore \frac{\mathrm{Total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{cylinder}}{\mathrm{Total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{cone}}=\frac{56\mathrm{\pi}{x}^{2}}{36\mathrm{\pi}{x}^{2}}=\frac{14}{9}$

Thus, the ratio of total surface area of the cylinder to the total surface area of the cone is 14 : 9.

A solid cylinder and a solid cone have equal bases and equal heights. If the radius and height be in the ratio 4 : 3, the ratio of the total surface area of the cylinder to that of the cone is _____14 : 9_____.

#### Page No 20.25:

#### Question 7:

If *h*, *S* and *V* be the height curved surface area and volume of a cone respectively, then 3π*Vh*^{3} – *S*^{2}*h*^{2 }+ 9*V*^{2} is equal to ________.

#### Answer:

Height of the cone = *h*

Let *r* be the radius and *l* be the slant height of the cone.

∴ *l*^{2} = *r*^{2} + *h*^{2} .....(1)

Volume of the cone =* V* = $\frac{1}{3}\mathrm{\pi}{r}^{2}h$

Curved surface area of the cone = *S* = $\mathrm{\pi}$*rl*

$\therefore 3\mathrm{\pi}V{h}^{3}\mathit{-}{S}^{2}{h}^{2}+9{V}^{2}$

$=3\mathrm{\pi}\times \frac{1}{3}\mathrm{\pi}{r}^{2}h\times {h}^{3}-{\mathrm{\pi}}^{2}{r}^{2}{l}^{2}\times {h}^{2}+9\times \frac{1}{9}{\mathrm{\pi}}^{2}{r}^{4}{h}^{2}$

$={\mathrm{\pi}}^{2}{r}^{2}{h}^{4}-{\mathrm{\pi}}^{2}{r}^{2}\left({r}^{2}+{h}^{2}\right)\times {h}^{2}+{\mathrm{\pi}}^{2}{r}^{4}{h}^{2}$ [Using (1)]

$={\mathrm{\pi}}^{2}{r}^{2}{h}^{4}-{\mathrm{\pi}}^{2}{r}^{2}{h}^{4}-{\mathrm{\pi}}^{2}{r}^{4}{h}^{2}+{\mathrm{\pi}}^{2}{r}^{4}{h}^{2}$

$=0$

Thus, the value of $3\mathrm{\pi}V{h}^{3}\mathit{-}{S}^{2}{h}^{2}+9{V}^{2}$ is 0.

If *h*, *S* and *V* be the height, curved surface area and volume of a cone respectively, then 3π*Vh*^{3} – *S*^{2}*h*^{2 }+ 9*V*^{2} is equal to _____0_____.

#### Page No 20.25:

#### Question 8:

If the heights of two cones are in the ratio 1 : 4 and the radii of their bases are in the ratio 4 : 1, then the ratio of their volumes is _________.

#### Answer:

Let *r* be the radius and *h* be the height of the first cone & *R* be the radius and *H* be the height of the second cone.

It is given that,

$\frac{h}{H}=\frac{1}{4}$ and $\frac{r}{R}=\frac{4}{1}$ .....(1)

Suppose *V*_{1} be the volume of the first cone and *V*_{2} be the volume of the second cone.

$\therefore \frac{{V}_{1}}{{V}_{2}}$

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

$={\left(\frac{r}{R}\right)}^{2}\times \frac{h}{H}$ [Using (1)]

$={\left(\frac{4}{1}\right)}^{2}\times \frac{1}{4}$

$=4$

Thus, the ratio of the volume of first cone to the volume of second cone is 4 : 1.

If the heights of two cones are in the ratio 1 : 4 and the radii of their bases are in the ratio 4 : 1, then the ratio of their volumes is _____4 : 1_____.

#### Page No 20.25:

#### Question 9:

The diameters of two cones are equal and their slant heights are in the ratio 5 : 4. If the curved surface area of the smaller cone is 200 cm^{2}, then the curved surface area of the bigger cone is _________.

#### Answer:

Let *l*_{1} be the slant height of first cone and *l*_{2} be the slant height of the second cone.

Given: *l*_{1} : *l*_{2} = 5 : 4

∴ *l*_{1} = 5*x* and *l*_{2} = 4*x*, where *x* is constant

It is given that the diameters of the two cones are equal. Therefore, the radii of the two cones are equal.

Let the radius of each cone be *r* cm.

Since the radii of two cones are equal, so the cone with bigger slant height would be bigger than the other.

Now,

Curved surface area of the bigger cone = *$\mathrm{\pi}$r**l*_{1}

Curved surface area of the smaller cone = *$\mathrm{\pi}$r**l*_{2}

$\therefore \frac{\mathrm{Curved}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{bigger}\mathrm{cone}}{\mathrm{Curved}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{smaller}\mathrm{cone}}=\frac{\mathrm{\pi}r{l}_{1}}{\mathrm{\pi}r{l}_{2}}$

$\Rightarrow \frac{\mathrm{Curved}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{bigger}\mathrm{cone}}{200{\mathrm{cm}}^{2}}=\frac{{l}_{1}}{{l}_{2}}=\frac{5x}{4x}$

⇒ Curved surface area of the bigger cone = $\frac{5}{4}\times 200{\mathrm{cm}}^{2}$ = 250 cm^{2}

Thus, the curved surface area of the bigger cone is 250 cm^{2}.

The diameters of two cones are equal and their slant heights are in the ratio 5 : 4. If the curved surface area of the smaller cone is 200 cm^{2}, then the curved surface area of the bigger cone is ______250 cm ^{2}______.

#### Page No 20.26:

#### Question 1:

The height of a cone is 15 cm. If its volume is 500$\mathrm{\pi}$ cm^{3}, then find the radius of its base.

#### Answer:

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

#### Page No 20.26:

#### Question 2:

If the volume of a right circular cone of height 9 cm is 48$\mathrm{\pi}$ cm^{3}, find the diameter of its base.

#### Answer:

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 cm^{3}

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

#### Page No 20.26:

#### Question 3:

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

#### Answer:

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 $2401\mathrm{\pi}{\mathrm{cm}}^{3}$.

#### Page No 20.26:

#### 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:

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

#### Page No 20.26:

#### Question 5:

If the radius and slant height of a cone are in the ratio 7 : 13 and its curved surface area is 286 cm^{2}, find its radius.

#### Answer:

It is given that the curved surface area (C.S.A) of the cone is 286 cm^{2}^{ }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* = 7*k*

*l* = 13*k*

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

Curved Surface Area, 286 =

286 = 286 *k*^{2}

1 = *k*^{2}

Hence the value of *k* = 1

From this we can find the value of base radius,

*r* = 7*k*

*r* = 7

Therefore the base radius of the cone is

#### Page No 20.26:

#### Question 6:

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

#### Answer:

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

*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

#### Page No 20.26:

#### 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:

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

#### Page No 20.26:

#### 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:

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

*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 m^{2}

It is given that the cloth is 100*π* 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

#### Page No 20.7:

#### Question 1:

Find the curved surface area of a cone, if its slant height is 60 cm and the radius of its base is 21 cm.

#### Answer:

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

Curved Surface Area =

Substituting the values of *r* = 21 cm and *l* = 60 cm in the above equation and using

Curved Surface Area will be,

=

=

=

Therefore the Curved Surface Area of the cone with the specified dimensions is .

#### Page No 20.7:

#### Question 2:

The radius of a cone is 5 cm and vertical height is 12 cm. Find the area of the curved surface.

#### Answer:

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

Curved Surface Area =

But, here we’re given only that the base radius *r* = 5 cm and vertical height *h* = 12 cm.

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

Slant height, *l*=

=

=

=

*l* = 13 cm

Now, substituting the values of *r* = 5 cm and slant height *l* = 13 cm and using in the formula of C.S.A,

We get Curved Surface Area =

=

=

Therefore the Curved Surface Area of the cone with the specified dimensions is .

#### Page No 20.7:

#### Question 3:

The radius of a cone is 7 cm and area of curved surface is 176 cm^{2}. Find the slant height.

#### Answer:

It is given that the curved surface area (C.S.A) of the cone is 176 cm^{2} and that the base radius is 7 cm. The formula of the curved surface area of a cone with base radius ‘*r*’ and slant height ‘l’ is given as

Curved Surface Area =

Hence, slant height, *l *=

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

Slant height, *l* =

= 8

Hence the slant height of the cone with the mentioned dimensions is.

#### Page No 20.7:

#### Question 4:

The height of a cone is 21 cm. Find the area of the base if the slant height is 28 cm.

#### Answer:

In a cone, the vertical height ‘*h*’ is given as 21 cm and the slant height ‘*l*’ is given as 28 cm, and the area of the base is asked. The base area is given as

Base area =

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.

Now, let us substitute the value of r in the formula for area of the base.

Base Area =

=

=

= 1078

Hence, the base area of the cone with the specified dimensions is.

#### Page No 20.7:

#### Question 5:

Find the total surface area of a right circular cone with radius 6 cm and height 8 cm.

#### Answer:

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

Total Surface Area =

But we do not have the slant height. We are given that *r* = 6 cm and* h *= 8 cm. We find *l* using the relation

=

=

=

= 10.

Therefore, the slant height, *l* = 10 cm.

Substituting the values of *r *= 6 cm and *l* = 10 cm in the above equation and using in specified formula,

Total Surface Area =

=

=

Therefore the total surface area of the given cone is or 301.71 cm^{2}.

#### Page No 20.7:

#### Question 6:

Find the curved surface area of a cone with base radius 5.25 cm and slant height 10 cm.

#### Answer:

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area = *πrl*

Substituting the values of *r *= 5.25 cm and *l* = 10 cm in the above equation and using

Curved Surface Area =

=

=

Therefore the Curved Surface Area of the cone with the specified dimensions is

#### Page No 20.8:

#### Question 7:

Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.

#### Answer:

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

Total Surface Area =

The diameter of the base is given as 24 m. The radius of the base is half of the diameter and hence *r* = 12 m.

Substituting the values of *r* = 12 m and *l* = 21 cm in the above equation and using in specified formula,

Total Surface Area =

=

=

Therefore the total surface area of the given cone is or 1244.57 m^{2}.

#### Page No 20.8:

#### Question 8:

The area of the curved surface of a cone is 60$\mathrm{\pi}$cm^{2}. If the slant height of the cone be 8 cm, find the radius of the base.

#### Answer:

It is given that the curved surface area (C.S.A) of the cone is cm^{2}^{ }and that the slant height is 8 cm. The formula of the curved surface area of a cone with base radius ‘*r*’ and slant height *‘l’* is given as

Curved Surface Area =

Hence, slant height, *r* =

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

Slant height, *r* =

= 7.5

Hence the base radius of the cone with the mentioned dimensions is.

#### Page No 20.8:

#### Question 9:

The curved surface area of a cone is 4070 cm^{2} and its diameter is 70 cm. What is its slant height? (Use $\mathrm{\pi}$ = 22/7).

#### Answer:

It is given that the curved surface area (C.S.A) of the cone is 4070 cm^{2} and that the base diameter is 70 cm. The formula of the curved surface area of a cone with base radius *‘r*’ and slant height ‘*l’* is given as

Curved Surface Area =

Hence, slant height, *l* =

The base radius is half of the base diameter. And since the base diameter is given as 70 cm we can find out the base radius as, *r* = 35 cm.

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

Slant height, *l* =

=

= 37

Hence the slant height of the cone with the mentioned dimensions is.

#### Page No 20.8:

#### Question 10:

The radius and slant height of a cone are in the ratio of 4 : 7. If its curved surface area is 792 cm^{2}, find its radius. (Use $\mathrm{\pi}$ = 22/7).

#### Answer:

It is given that the curved surface area (C.S.A) of the cone is 792 cm^{2} and that the ratio between the base radius and the slant height is 4: 7. The formula of the curved surface area of a cone with base radius ‘*r*’ and slant height ‘*l*’ is given as

Curved Surface Area =

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* = 4*k*

*l* = 7*k*

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

Curved Surface Area,

792 =

792 =

9 =

Hence the value of *k* = 3.

From this we can find the value of base radius,

*r* = 4*k*

*r* = 12

Therefore the base radius of the cone is.

#### Page No 20.8:

#### Question 11:

A joker's cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps.

#### Answer:

The area of sheet required to make a cone would be equal to the curved surface area of the cone that is to be formed. So here we need to find the C.S.A. of a single cone and then multiply the same to arrive at the final answer.

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area =

But, here we’re given only that the base radius *r* = 7 cm and vertical height *h* = 24 cm.

*l*’ to be used in the formula for Curved Surface Area we use the following relation

Slant height, *l* =

=

=

=

*l* = 25 cm

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

We get Curved Surface Area =

=

Thus the curved surface area of one cone is 550 cm^{2}. Since we require 10 such joker cap the required sheet area would be 10 times this value.

Hence the area of sheet required to make 10 joker caps of the specified dimensions would be

#### Page No 20.8:

#### Question 12:

Find the ratio of the curved surface areas of two cones if their diameters of the bases are equal and slant heights are in the ratio 4 : 3.

#### Answer:

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 there are two cones with base radius and slant heights as, and, respectively.

Since the base diameters of both the cones are equal we get that = =

Since only the ratio between slant heights of the two cones is given as 4: 3, we shall use them by introducing a constant ‘*k*’

So, now = 4*k*

= 3*k*

Using these values we shall evaluate the ratio between the curved surface areas of the two cones

=

=

=

Hence the ratio between the curved surface areas of the two cones with the mentioned dimensions is

#### Page No 20.8:

#### Question 13:

There are two cones. The curved surface area of one is twice that of the other. The slant height of the later is twice that of the former. Find the ratio of their radii.

#### Answer:

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area = *πrl*

Now there are two cones with base radius, slant height and Curved Surface Area (C.S.A) as,, , , respectively.

It is given that = 2() and also that = 2(). Or this can also be written as

=

=

=

=

=

Therefore the ratio between the base radiuses of the two cones is

#### Page No 20.8:

#### Question 14:

The diameters of two cones are equal. If their slant heights are in the ratio 5 : 4, find the ratio of their curved surfaces.

#### Answer:

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area = *πrl*

Now there are two cones with base radius and slant heights as, and, respectively.

Since the base diameters of both the cones are equal we get that = =

Since only the ratio between slant heights of the two cones is given as 5: 4, we shall use them by introducing a constant ‘*k’*

So, now

= 5*k*

= 4*k*

Using these values we shall evaluate the ratio between the curved surface areas of the two cones

=

=

=

Hence the ratio between the curved surface areas of the two cones with the mentioned dimensions is

#### Page No 20.8:

#### Question 15:

Curved surface area of a cone is 308 cm^{2} and its slant height is 14 cm. Find the radius of the base and total surface area of the cone.

#### Answer:

It is given that the curved surface area (C.S.A) of the cone is 308 cm^{2} and that the slant height is 14 cm. 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*

Hence, base radius, *r* =

Substituting the values of C.S.A and the slant height and using in the above equation we get

*r* =

*r* = 7

Hence the value of the base radius is

*r*’ and slant height ‘*l*’ is given as

Total Surface Area =

Substituting the values of *r* = 7 m and *l* = 14 cm in the above equation and using in specified formula,

Total Surface Area =

= (22) (21)

=

Therefore the total surface area of the given cone is

#### Page No 20.8:

#### Question 16:

The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of Rs 210 per 100 m^{2}.

#### Answer:

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area = *πrl*

The base diameter is given as 14 m. Hence the base radius, *r* = 7 m.

Substituting the values of *r* = 7 m and *l* = 25 m in the above equation and using

Curved Surface Area =

=

=

The curved surface area of the conical tomb to be white-washed is 550 m^{2}

The cost of white washing is given as Rs. 210 per 100 m^{2}

This works out to Rs. 2.10 per m^{2}

Total cost (T.C) of white washing the conical tomb is

T.C. = (Total area to be white-washed) (Cost per m^{2})

= (550) (2.10)

= 1155

So the total cost of white-washing the given curved surface area is

#### Page No 20.8:

#### Question 17:

A conical tent is 10 m high and the radius of its base is 24 m. Find the slant height of the tent. If the cost of 1 m^{2} canvas is Rs 70, find the cost of the canvas required to make the tent.

#### Answer:

It is given that the vertical height ‘*h*’ = 10 m and base radius ‘*r*’ = 24 m.

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

Slant height,

*l* =

=

=

=

*l* = 26 m

Hence the slant height of the given cone is

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

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area =

Now, substituting the values of *r* = 24 m and slant height *l* = 26 m and using in the formula of C.S.A,

We get Curved Surface Area =

=

Therefore the Curved Surface Area of the cone is m^{2}

The cost of the canvas is given as Rs. 70 per m^{2}

The total cost of canvas= (Total curved surface area) (Cost per m^{2})

= (70)

= 137280

Hence the total amount required to construct the tent is

#### Page No 20.8:

#### Question 18:

A tent is in the form of a right circular cylinder surmounted by a cone. The diameter of cylinder is 24 m. The height of the cylindrical portion is 11 m while the vertex of the cone is 16 m above the ground. Find the area of the canvas required for the tent.

#### Answer:

The tent being in the form of a cone surmounted on a cylinder the total amount of canvas required would be equal to the sum of the curved surface areas of the cone and the cylinder.

The diameter of the cylinder is given as 24 m. Hence its radius, *r* = 12 m. The height of the cylinder, *h* = 11 m.

The curved surface area of a cylinder with radius ‘*r*’ and height ‘*h*’ is given by the formula

Curved Surface Area of the cylinder =

Substituting the values of *r* = 12 m and *h* = 11 m in the above equation

Curved Surface Area of the cylinder =

=

The vertex of the cone is given to be 16 m above the ground and the cone is surmounted on a cylinder of height 11 m, hence the vertical height of the cone is *h* = 5 m. The radius of the cone is the same as the radius of the cylinder and so base radius, *r* = 12 m.

*l*’ to be used in the formula for Curved Surface Area we use the following relation

Slant height, *l* =

=

=

=

*l* = 13 m

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area =

Substituting the values of *r* = 12 m and *l* = 13 m in the above equation

We get

Curved Surface Area of the cone =

=

Total curved surface area = Curved surface area of cone + curved surface area of cylinder

= +

=

=

= 1320

Thus the total area of canvas required is

#### Page No 20.8:

#### Question 19:

What length of tarpaulin 3 m wide will be required to make a conical tent of height 8 m and base radius 6 m? Assume that the extra length of material will be required for stitching margins and wastage in cutting is approximately 20 cm (Use $\mathrm{\pi}$ = 3.14)

#### Answer:

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

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area = *πrl*

It is given that the base radius *r* = 6 m and vertical height *h* = 8 m.

*l*’ to be used in the formula for Curved Surface Area we use the following relation

Slant height,

*l* =

=

=

=

l = 10 m

Now, substituting the values of *r* = 6 m and slant height *l* = 10 m and using *π* = 3.14 in the formula of C.S.A,

We get Curved Surface Area =

= 188.4

Hence the curved surface area of the cone is 188.4 m^{2}

Now, the width of the canvas is 3 m.

Area of the canvas required = (Width of the canvas) (Length of the canvas)

Therefore,

Length of the canvas =

=

= 62.8

Length of canvas is 62.8 m. But we need to add another 20 cm of length for wastage.

20 cm = 0.2 m.

Hence the total amount of canvas length required is

#### Page No 20.8:

#### Question 20:

A bus stop is barricated from the remaining part of the road, by using 50 hollow cones made of recycled card-board. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is Rs 12 per m^{2}. What will be the cost of painting all these cones. (Use $\mathrm{\pi}=3.14$ and $\sqrt{1.04}=1.02)$

#### Answer:

The area to be painted is the curved surface area of each cone.

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area = *πrl*

For each cone, we’re given that the base diameter is 0.40 m.

Hence the base radius *r* = 0.20 m. The vertical height *h* = 1 m.

*l*’ to be used in the formula for Curved Surface Area we use the following relation

Slant height,

*l* =

=

=

=

*l* = 1.02 m

Now, substituting the values of *r* = 0.2 m and slant height *l* = 1.02 m and using *π* = 3.14 in the formula of C.S.A,

We get Curved Surface Area =

= 0.64056 m^{2}

This is the curved surface area of a single cone. Since we need to paint 50 such cones the total area to be painted is,

Total area to be painted = (0.64056) (50)

= 32.028 m^{2}

The cost of painting is given as Rs. 12 per m^{2}

Hence the total cost of painting = (12) (32.028)

= 384.336

Hence, the total cost that would be incurred in painting is

#### Page No 20.8:

#### Question 21:

A cylinder and a cone have equal radii of these bases and equal heights. If their curved surface areas are in the ratio 8:5, show that the radius of each is to the height of each as 3:4.

#### Answer:

It is given that the base radius and the height of the cone and the cylinder are the same.

So let the base radius of each is ‘*r*’ and the vertical height of each is ‘*h*’.

Let the slant height of the cone be ‘*l*’

The curved surface area of the cone =

The curved surface area of the cylinder =

It is said that the ratio of the curved surface areas of the cylinder to that of the cone is 8:5

So,

=

=

=

But we know that *l* =

=

Squaring on both sides we get

=

=

=

= – 1

=

=

Hence it is shown that the ratio of the radius to the height of the cone as well as the cylinder is

#### Page No 20.8:

#### Question 22:

The circumference of the base of a 10 m height conical tent is 44 metres. Calculate the length of canvas used in making the tent if width of canvas is 2 m. (Use $\mathrm{\pi}$ = 22/7).

#### Answer:

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

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area = *πrl*

It is given that the circumference of the base is 44 m.

So,

= 10

=

= 7 m

It is given that the vertical height of the cone is *h* = 10 m.

*l*’ to be used in the formula for Curved Surface Area we use the following relation

Slant height,

*l* =

=

=

=

*l* = m

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

We get Curved Surface Area =

=

Hence the curved surface area of the given cone is m^{2}

Now, the width of the canvas is 5 m.

Area of the canvas required = (Width of the canvas) (Length of the canvas)

Therefore,

Length of the canvas =

=

= 134.27

Hence the length of canvas required is

#### Page No 20.8:

#### Question 23:

A circus tent is cylindrical to a height of 3 metres and conical above it. If its diameter is 105 m and the slant height of the conical portion is 53 m, calculate the length of the canvas 5 m wide to make the required tent.

#### Answer:

We need to find out the total amount of canvas required to make the circus tent. The height of the cylindrical portion is given as *h* = 3 m, and the diameter is given as 105 m.

Hence the radius *r* = m.

The curved surface area of a cylinder with radius ‘*r*’ and height ‘*h*’ is given by the formula

Curved Surface Area of the cylinder =

Substituting the values of *r* = m and *h* = 3 m in the above equation

Curved Surface Area of the cylinder =

=

= (22) (15) (3)

= 990

Hence the curved surface area of the cylinder is 990 m^{2}

The slant height of the cone is *l* = 53 m. The base radius of the cone is the same as the radius of the cylinder and hence *r* =

*r*’ and slant height ‘*l*’ is given as

Curved Surface Area = *πrl*

Substituting the values of *r* = m and *l* = 53 m in the above equation

We get

Curved Surface Area of the cone =

= 8745

Hence the curved surface area of the cone is 8745 m^{2}

Total curved surface area = Curved surface area of cone + curved surface area of cylinder

= 8745 + 990

= 9735

The total surface area of the tent is 9735 m^{2}

Now, the width (or) breadth of the canvas is 5 m.

Area of the canvas required = (Breadth of the canvas) (Length of the canvas)

Therefore,

Length of the canvas =

=

= 1947

Hence the length of canvas required is

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