Chapter 20: Surface Area And Volume of A Right Circular Cone Exercise – 20.2

Question: 1

Find the volume of the right circular cone with the following dimensions:

(a) Radius is 6 cm and the height of the cone is 7cm

(b) Radius is 3.5 cm and height is 12 cm

(c) Slant height is 21 cm and height is 28 cm

Solution:

(a) It is given that        

Radius of the cone (r) = 6 cm

Height of the cone (h) = 7 cm

Volume of a right circular cone

= 1/3πr²h

= 1/3 ∗ 3.14 ∗ 6² ∗ 7 = 264 cm³

(b) It is given that:

Radius of the cone (r) = 3.5 cm

Height of the cone (h) = 12 cm

Volume of a right circular cone

= 1/3πr²h

= 1/3 ∗ 3.14 ∗ 3.5² ∗ 12

= 154 cm³

(c) It is given that:

Height of the cone (h) = 28 cm

Slant height of the cone (l) = 21 cm

As we know that,

l² = r² + h²

Volume of a right circular cone:

= 1/3πr²h

= 7546 cm³

 

Question: 2

Find the volume of a conical tank with the following dimensions in liters:

(a) Radius is 7cm and the slant height of the cone is 25 cm

(b) Slant height is 12cm and height is 13 cm

Solution:

(a) It is given that

Radius of the cone(r) = 7 cm

Slant height of the cone (l) = 25 cm

As we know that,

l² = r² + h²

= 24 cm

Volume of a right circular cone

= 1/3 πr²h

= 1/3 ∗ 3.14 ∗ 7² ∗ 24

= 1232 cm³ = 1.232 litres [1 cm³ = 0.01l]

(c) It is given that:

Height of the cone (h) = 12 cm

Slant height of the cone (l) = 13 cm

As we know that,

l² = r² + h²

= 5 cm

Volume of a right circular cone:

= 1/3 πr²h

= 1/3 ∗ 3.14 ∗ 5² ∗ 10 = 314.85 cm³ = 0.307 litres   [1 cm³ = 0.01l]

 

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.

Solution:

Let ratio of the height of the cone be h_x

height of the 1^st cone = h_x

height of the of the  2^nd cone =3h_x

Let the ratio of the radius of the of the cone = r_x

radius of the 1^st cone = 3 r_x

radius of the 2^nd cone = r_x

The ratio of the volume = v₁/v₂

Where v₁ = volume of 1^st cone

v₂ = volume of 2^nd cone

 

Question: 4

The radius and the height of a right circular cone are in the ratio 5:12. If its volume is 314 cubic meter, find the slant height and the radius. (Use π = 3.14).

Solution:

Let us assume the ratio to be y

Radius (r) = 5y

Height (h) = 12y

We know that

l² = r² + h²

= 5y² + 12y²

= 25y² + 144y²

= 169² = 13y

Now it is given that volume = 314 m³

⟹ 1/3πr²h = 314 m³

⟹ 1/3 ∗ 3.14 ∗ 25y² ∗ 12y = 314 m³

⟹ y³ = 1

⟹ y = 1

Therefore,

Slant height (l) = 13y = 13 m

Radius = 5y = 5 m

 

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 π = 3.14).

Solution:

Let the ratio be y

The radius of the cone(r) = 5y

Height of the cone = 12y

Now we know,

= 13y

Now the volume of the cone is given 2512 cm³

⟹ 1/3 πr²h = 2512

⟹ 1/3 ∗ 3.14 ∗ 5y² ∗ 12y = 2512

⟹ y = 2

Therefore,

Slant height (l) = 13y = 13 * 2 = 26 cm

Radius of cone = 5y = 5 * 2 = 10 cm

 

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.

Solution:

Let the ratio of the radius be x

Radius of 1^st cone = 2x

Radius of 2^nd cone = 3x

Let the ratio of the volume be y

Volume of 1^st cone = 4y

Volume of 2^nd cone = 5y

y₁/y₂ = 4y/5y

= 4/5

Therefore the heights are in the ratio of 9:5.

 

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.

Solution:

It's given that

A cylinder and a cone are having equal radii of their bases and heights

Let the radius of the cone =radius of the cylinder = r

Height of the cone=height of the cylinder = h

Let the volume of cone = v_x

Volume of cylinder = v_y

⟹ v_y/v_x = 3/1

Therefore the ratio of their volumes is 3:1.

 

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?

Solution:

Let the radius of the cone be r_x and height be h_x

Then the volume of cone

Now,

Radius of the reduced cone = r_x/2

Therefore volume of reduced cone vy

Therefore the ratio between the volumes of the reduced and the original cone is 1: 4.

 

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 is canvas cloth required to just cover the heap?  (Use π = 3.14).

Solution:

It is given that

Diameter of heap (d) = 9 m

Therefore, Radius of the heap (r)

= d/2

= 9/2 = 4.5 m

Height of the heap (h) = 3.5 m

Therefore, Volume of the heap = 1/3πr²h

= 1/3 ∗ 3.14 ∗ 4.5² ∗ 3.5

= 74.18 m³

Now,

Area to be covered by the cloth = Curved surface area of the heap

= πrl = 3.14 * 4.5 * 5.70 = 80.54 m³

 

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.

Solution:

It is given that:

Diameter (d) = 14 cm

Height of the cone (h) = 51cm

Radius of the cone(r) = d/2

= 14/2 = 7cm

Therefore, Volume of cone (v) = 1/3πr²h

= 1/3 * 3.14 * 7 * 5 * 51 = 2618 cm³

Now it is given that 1 cm³ material weighs 10 gm.

Therefore, 2618 cm³ weighs = 2618 * 10 = 26180 grams.

 

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.

Solution:

It is given that

Radius of cone(r) = 6.3 cm

Height of the cone (h) = 10 cm

We know that

Therefore Volume of cone (v) = 1/3πr²h

= 1/3 ∗ 3.14 ∗ 6.3² ∗ 10 = 415.8 cm³

Curved surface area of cone = πrl

= 3.14 * 6.3 * 11.819 = 234.01cm²

 

Question: 12

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

Solution:

Radius of the base of the largest cone = 1/2 * edge of the cube

= 1/2 *14 = 7 cm

Height of the cone = Edge of the cube = 14 cm

Therefore, Volume of cone (v) = 1/3πr²h

= 1/3 ∗ 3.14 ∗ 7² ∗ 14 = 718.66 cm³

 

Question: 13

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

(a) Height of the cone

(b) Slant height of the cone

(c) Curved surface area of the cone

Solution:

(a) It is given that diameter of the cone (d) = 28 cm

Radius of the cone(r) = d/2

= 28/2 = 14cm

Height of the cone = ?

Now,

Volume of the cone (v) = 1/3πr²h = 9856 cm³

⟹ 1/3 ∗ 3.14 ∗ 14² ∗ h = 9856

Therefore the height of the cone is 48 cm

(b) It is given that

Radius of the cone(r) = 14 cm

Height of the cone = 48 cm

Slant height (l) = ?

Now we know that 

Therefore the slant height of the cone is 50 cm.

(c) Radius of the cone(r) = 14 cm

Slant height of the cone (l) = 50 cm

Curved surface area (C.S.A) = ?

Curved surface area of a cone (C.S.A) = πrl

= 3.14 * 14 * 50 = 2200 cm²

Therefore curved surface of the cone is 2200 cm²

 

Question: 14

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

Solution:

It is given that:

Diameter of the conical pit (d) = 3.5 m

Height of the conical pit (h) = 12 m

Radius of the conical pit (r) = ?

Volume of the conical pit (v) = ?

Radius of the conical pit(r) = d/2 = 3.5/2 = 1.75 m

Volume of the cone (v) = 1/3πr²h

= 1/3 ∗ 3.14 ∗ 1.75² ∗ 12 = 38.5 m³

Capacity of the pit = (38.5 * 1) kilolitres = 38.5 kilolitres

 

Question: 15

Monica has a piece of Canvas whose area is 551m². She uses it to have a conical tent made, with a base radius of 7m. 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.

Solution:

It is given that:

Area of the canvas = 551m²

Area that is wasted = 1m²

Radius of tent = 7m, Volume of tent (v) = ?

Therefore the Area of available for making the tent = (551 - 1) = 550 m²

Surface area of tent = 550 m²

⟹ πrl = 550

⟹ l = 550/22 = 25 m

Slant height (l) = 25 m

We know that,

l² = r² + h²

25² = 7² + h²

⟹ 625 - 49 = h²

⟹ 576 = h²

h = 24 m

Height of the tent is 24 m.

Now, volume of cone = 1/3πr²h

= 1/3 ∗ 3.14 ∗ 7² ∗ 24 = 1232 m³

Therefore the volume of the conical tent is 1232 m³.