Question 1

The height of cylinder is 14 cm and its curved surface area is 264 sq. cm. The volume of the cylinder is?

Accepted Answer

∵ 2πrh = 264 ⇒ 2 x (22/7) x r x 14 = 264 ∴ r = 3 So, Volume = πr2h = (22/7) x 3 x 3 x 14 cm3 = 396 cm3

Question 2

The percentage increased in the surface area of a cube when each side is doubled, is?

Accepted Answer

Original area = 6a2 New area = 6(2a)2 = 24a2 Required increased % = (18a2/6a2) x 100% = 300%

Question 3

If the radius of a sphere is doubled, then its surface area is increased by?

Accepted Answer

Original area = 4π r2, New area = 4π(2r)2 = 16 π r2 Required increased % = [12πr2]/[4π r2] x 100 %= 300%

Question 4

If the height of a come is doubled, then its volume is increased by?

Accepted Answer

Original volume = 1/3 πr2h; New volume = 1/3 πr2(2h) = 2/3πr2h Required increased % = [πr2h/3]/[πr2h/3] x 100% = 100%

Question 5

If the radius of a sphere is doubled, then its volume is increased by?

Accepted Answer

Original volume = 4/3πr3 New volume = 4/3 π (2r)3 = 32/3 πr3 Required increased % = [(28/3) πr3] x [(3/4)πr3] x 100 % = 700 %

Question 6

The radius of two cylinders are in the ratio of 2 : 3 and their height are in the ratio 5 : 3 . The ratio of their volumes is?

Accepted Answer

Let their radii be 2r and 3r and height 5h and 3h respectively. ∴ Ratio of their volumes = [π(2r)2 x 5h] / [π(3r)2 x 3h] = 20/27 = 20 : 27

Question 7

Two circular cylinders of equals volume have their heights in the ratio 1 : 2 Ratio of their radii is?

Accepted Answer

Let their height be h and 2h radii be x and y respectively. Then, πx2h = πy2(2h) ⇒ x2/y2 = 2/1 ∴ x/y = √2 / 1 = √2 : 1

Question 8

The radii of two spheres are in the ratio 1 : 2 . The ratio of their surface areas is?

Accepted Answer

Let their radii be x and 2x . Ratio of their surface areas = [4πx2]/[4π(2x)2] = 1/4 = 1 : 4

Question 9

A radius of the base of a right circular cone is 6 cm and its slant height is 28 cm. The curved surface area of the cone is?

Accepted Answer

Curved surface area = πrl = (22/7 x 6 x 28) cm2 = 528 cm2

Question 10

How many bullets can be made out of a cube of lead whose edge measures 22 cm, each bullet being 2 cm in diameter?

Accepted Answer

Number of bullets = Volume of cube / Volume of 1 bullet = (22 x 22 x 22)/(4/3 x 22/7 x 1 x 1 x 1) = 2541.

Question 11

A spherical lead ball of radius 10 cm is melted and small lead balls of radius 5 mm are made. The total number of possible small lead balls.?

Accepted Answer

Number of balls = Volume of big ball / Volume of 1 small ball = [4/3 x π x 10 x 10 x 10] / [4/3 x π x 0.5 x 0.5 x 0.5] = 8000

Question 12

A come of height 7 cm and base radius 3 cm is curved from a rectangular block of wood 10 cm x 5 cm x 2 cm. The percentage of wood wasted is?

Accepted Answer

Total volume of cuboid = (10 x 5 x 2) cm3 = 100 cm3 Volume curved = 1/3 x (22/7) x 3 x 3 x 7 cm3 = 66 cm3 % of wood wasted = (100 - 66)% = 34 %

Question 13

The material of a cone is converted into the shape of a cylinder of equal radius . If the height of the cylinder is 5 cm. the height of the cone is?

Accepted Answer

∵ (1/3)πr2 x h = πr2 x 5 ∴ h = 15 cm

Question 14

A cylindrical piece of metal of radius 2 cm and height 6 cm is shaped into a cone of same radius. The height of the cone is?

Accepted Answer

∴ 1/3 π x (2)2 x h = π x (2)2 x 6 ⇒ h = 18 cm

Question 15

If the volumes of two cones are in ratio 1 : 4 and their diameter are in the ratio 4 : 5, then the ratio of their heights is?

Accepted Answer

Since the diameters are in the ratio 4 : 5. It follows that their radii are in the ratio 4 : 5. Let them be 4r and 5r. Let the height be h and H . ∵ Ratio of volumes = [(1/3)π(4r)2h / (1/3)π(5r)2H = (16h)/(25H) ⇒ h/H = (1/4) x (25/16) = 25/64 = 25: 64.

Question 16

If a right circular cone of vertical height 24 cm has a volume of 1232 cm 3, then the area of its curved surface is?

Accepted Answer

∵ 1/3 x (22/7) x r2 x 24 = 1232 ⇒ r2 = 1232 x (7/22) x (3/24) = 49 ∴ r = 7 cm Now, h = 24 So, l = √72 + (24)2 = √625 = 25 cm ∴ Curved surface area = πrl = (22/7) x 7 x 25 cm2 = 550 cm2

Question 17

A cylindrical vessel 60 cm in diameter is partially filled with water. A sphere 60 cm in diameter is gently dropped into the vessel. To what further height will water rise in the cylinder?

Accepted Answer

Let h and H be the height of water level before and after the dropping of the sphere. Then, [π x (30)2 x H ] - [π x (30)2 x h ] = 4/3 π x (30)3 ⇒ π x 900 x (H -h) = 4/3 π x 27000 ∴ (H- h) = 40 cm

Question 18

The area of the base of a right circular cone is 154 cm 2 and its height 14 cm. The curved surface of the cone is?

Accepted Answer

∵ 22/7 x r2 = 154 ⇒ r2 = 154 x (7/22) = 49 ∴ r = 7 cm and h = 14 So, l = √(7)2 + (14)2 = √245 = 7√5 cm ∴ Area of curved surface = πrl = (22/7) x 7 x 7√5 cm2 = 154√5 cm2

Question 19

Volume of a hemisphere is 19404 cubic cm. The total surface area is?

Accepted Answer

(2/3) x (22/7) x r3 = 19404 ⇒ r3 = 19404 x (7/22) x (3/2) = 9261 = (21)3 ∴ r = 21 cm Total Surface area = 3πr2 = 3 x (22/7) x 21 x 21 cm2 = 4158 cm2

Question 20

A spherical ball of lead, 3 cm in diameter is melted and recast into three spherical balls, The diameter of two of these are 1.5 cm and 2 cm respectively. The diameter of the third ball is?

Accepted Answer

∵ 4/3πr3 = 4/3π x [(3/2)3 - {(3/4)3 + 13 } ] ⇒ r3 = 125/64 = (5/4)3 ⇒ r = 5/4 ∴ Diameter = (5/4) x 2 cm = 2.5 cm.

Volume and Surface Area Questions