RD Sharma 2020 solution class 9 chapter 20 Surface Area and Volume of a Right Circular Cone Exercise 20.1

Exercise 20.1

Page-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 1:

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 .

Question 2:

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

Answer 2:

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 .

Question 3:

The radius of a cone is 7 cm and area of curved surface is 176 cm². Find the slant height.

Answer 3:

It is given that the curved surface area (C.S.A) of the cone is 176 cm² 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.

Question 4:

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

Answer 4:

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.

Question 5:

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

Answer 5:

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 cm².

Question 6:

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

Answer 6:

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

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-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 7:

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 m².

Question 8:

The area of the curved surface of a cone is 60$\mathrm{\pi }$cm². If the slant height of the cone be 8 cm, find the radius of the base.

Answer 8:

It is given that the curved surface area (C.S.A) of the cone is cm² 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.

Question 9:

The curved surface area of a cone is 4070 cm² and its diameter is 70 cm. What is its slant height? (Use $\mathrm{\pi }$ = 22/7).

Answer 9:

It is given that the curved surface area (C.S.A) of the cone is 4070 cm² 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.

Question 10:

The radius and slant height of a cone are in the ratio of 4 : 7. If its curved surface area is 792 cm², find its radius. (Use $\mathrm{\pi }$ = 22/7).

Answer 10:

It is given that the curved surface area (C.S.A) of the cone is 792 cm² 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 = 4k

l = 7k

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 = 4k

r = 12

Therefore the base radius of the cone is.

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 11:

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.

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 = 7 cm and vertical height h = 24 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 = 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 cm². 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

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 12:

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 = 4k

= 3k

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

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 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

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

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 14:

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

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

= 5k

= 4k

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

Question 15:

Curved surface area of a cone is 308 cm² and its slant height is 14 cm. Find the radius of the base and total surface area of the cone.

Answer 15:

It is given that the curved surface area (C.S.A) of the cone is 308 cm² 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

The formula of the total surface area of a cone with base radius ‘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

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 m².

Answer 16:

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 

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 m²

The cost of white washing is given as Rs. 210 per 100 m²

This works out to Rs. 2.10 per m²

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

T.C. = (Total area to be white-washed) (Cost per m²)

= (550) (2.10) 

= 1155

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

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 m² canvas is Rs 70, find the cost of the canvas required to make the tent.

Answer 17:

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.

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 = 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 m²

The cost of the canvas is given as Rs. 70 per m²

The total cost of canvas= (Total curved surface area) (Cost per m²)

= (70)

= 137280

Hence the total amount required to construct the tent is

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 18:

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.

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 m

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 = 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

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 19:

The total amount of canvas required 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 base radius r = 6 m and vertical height h = 8 m.

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 = 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 m²

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

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 m². What will be the cost of painting all these cones. (Use $\mathrm{\pi } = 3.14$ and $\sqrt{1.04}=1.02)$

Answer 20:

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

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.

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 = 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 m²

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 m²

The cost of painting is given as Rs. 12 per m²

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

= 384.336

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

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 21:

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

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 22:

The total amount of canvas required 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 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.

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 = 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 m²

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

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 23:

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 m²

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 =

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 

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 m²

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 m²

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