## Chapter 13 Surface Areas and Volumes Exercise 13.7

Question 1: Find the volume of the right circular cone with

i) radius 6 cm, height 7 cm

ii) radius 3.5 cm, height 12 cm

Answer:

i) Given

Radius of the cone = 6 cm

Height of the cone = 7 cm

Volume of the cone = ?

Volume of the cone = 1/3πr²h

= 1/3 × 22/7 × 6 × 6 × 7

= 22 × 2 × 6

= 264 cm³

ii) Given

Radius of the cone = 3.5 cm

Height of the cone = 12 cm

Volume of the cone = ?

Volume of the cone = 1/3πr²h

= 1/3 × 22/7 × 3.5 × 3.5 × 12

= 22 × 0.5 × 3.5 × 4

= 154 cm³

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:

i) Given

Radius of the cone = 7 cm

Slant height of the cone = 25 cm

Height of the cone = √25² - 7² = √625 - 49 = √576 = 24 cm

Volume of cone = 1/3πr²h

= 1/3 × 22/7 × 7 × 7 × 24

= 22 × 7 × 8

= 1232 cm³

1 l = 1000 cm³

So, 1232 cm³ = 1232/1000 = 1.232 l

ii) Given

Height of the cone = 12 cm

Slant height of the cone = 13 cm

Radius of the cone = √13² - 12² = √169 - 144 = √25 = 5 cm

Volume of cone = 1/3πr²h

= 1/3 × 22/7 × 5 × 5 × 12

= 22/7 × 5 × 5 × 4

= 2200/7 cm³

= 314.28 cm³

1 l = 1000 cm³

So, 314.28 cm³ = 314.28/1000 = 0.31428 l

Question 3: The height of a cone is 15 cm. If its volume is 1570 cm³, find the radius of the base. (Use π = 3.14)

Answer:

Given

Height of the cone = 15 cm

Volume of the cone = 1570 cm³

Radius of the base of the cone =?

Volume of cone = 1/3πr²h

1570 = 1/3 × 3.14 × r² × 15

1570 = 15.7r²

r² = 1570/15.7

r² = 100

r = √100

r = 10 cm

Therefore, the radius of the base of the cone is 10 cm.

Question 4: If the volume of a right circular cone of height 9 cm is 48π cm³, find the diameter of its base.

Answer:

Given

Volume of cone = 48π cm³

Height of the cone = 9 cm

Diameter of the base = ?

Volume of the cone = 1/3πr²h

48π = 1/3× π × r² × 9

48π = 3πr²

r² = 48π/3π

r² = 16

r = √16

r = 4 cm

Diameter = 2 × Radius = 2 × 4 = 8 cm

Therefore, the diameter of the base of the cone is 8 cm.

Question 5: A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kilolitres?

Answer:

Given

Diameter of the conical pit = 3.5 m

Height of the conical pit = 12 m

Capacity in kilolitres = ?

Radius = Diameter/2 = 3.5/2 m

Volume of cone = 1/3πr²h

= 1/3 × 22/7 × 3.5/2 × 3.5/2 × 12

= 11 × 0.5 × 3.5 × 2

= 38.5 m³

1 kilolitre = 1 m³

38.5 m³ = 38.5 kl

Therefore, the capacity of the conical pit in kilolitres is 38.5 kl.

Question 6: The volume of a right circular cone is 9856 cm³. If the diameter of the base is 28 cm, find

i) height of the cone

ii) slant height of the cone

iii) curved surface area of the cone

Answer:

Given

Volume of the cone = 9856 cm³

Diameter of the base of the cone = 28 cm

Height of the cone = ?

Slant height of the cone = ?

Curved Surface Area of the cone = ?

Radius = Diameter/2 = 28/2 = 14 cm

Volume of the cone = 1/3πr²h

9856 = 1/3 × 22/7 × 14 × 14 × h

9856 = 1/3 × 22 × 2 × 14 × h

9856 = 616h/3

h = 9856 × 3/616

h = 48 cm

Slant height of cone = √r² + h²

= √(14)² + (48)²

= √196 + 2304

= √2500

= 50 cm

Curved Surface Area of cone = πrl

= 22/7 × 14 × 50

= 2200 cm²

Therefore, the height of the cone is 48 cm, slant height of the cone is 50 cm and the CSA of the cone is 2200 cm².

Question 7: A right triangle ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12 cm. Find the volume of the solid so obtained.

Answer:

If a right angled triangle is revolved, then a cone is obtained.

Given

Radius of the cone = 5 cm

Height of the cone = 12 cm

Volume of the cone = 1/3πr²h

= 1/3 × π × 5 × 5 × 12

= 5 × 5 × 4 × π

= 100π cm³

Question 8: If the triangle ABC in the Question 7 above is revolved about the side 5 cm, then find the volume of the solid so obtained. Find also the ratio of the volumes of the two solids obtained in Questions 7 and 8.

Answer:

Given

Radius of the cone = 12 cm

Height of the cone = 5 cm

Volume of the cone = 1/3πr²h

= 1/3 × π × 12 × 12 × 5

= 5 × 12 × 4 × π

= 240π cm³

Ratio of Volume of cone (Question 7)/Volume of cone (Question 8)

= 100π/240π

= 5/12

= 5:12

Question 9: A heap of wheat is in the form of a cone whose diameter is 10.5 m and height is 3 m. Find its volume. The heap is to be covered by canvas to protect it from rain. Find the area of the canvas required.

Answer:

Given

Height of the heap = 3 m

Diameter of the heap = 10.5 m

Radius = Diameter/2 = 10.5/2 m

l = √r² + h²

l = √(5.25)²+ (3)²

l = √27.5625 + 9

l = √36.5625

l = 6.04 m (approx)

Volume of the cone = 1/3πr²h

= 1/3 × 22/7 × 10.5/2 × 10.5/2 × 3

= 11 × 1.5 × 10.5/2

= 173.25

= 86.625 m³

Area of the canvas required to cover the heap of wheat = πrl

= 22/7 × 10.5/2 × 6.04

= 11 × 1.5 × 6.04

= 99.66 m²

Therefore, the volume of the heap is 86.625 m³ and the area of the canvas required is 99.66 m².

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