Question 1

A wooden box (open at the top) of thickness 0.5 cm, length 21 cm, width 11 cm and height 6 cm is painted on the inside. The expenses of painting are ₹ 70. What is the rate of painting per square centimetre?

Accepted Answer

As the box is painted from inside, the dimensions of the box from inside are Length = (21 - 05 - 05) = 20 cm Breadth = (11 - 05 - 05) = 10 cm and height = (6 - 05) = 5.5 Total number of faced to be painted = 4 walls + One base The dimensions of two of the walls = 2 x (10 x 5.5) The dimensions of the remaining two = 2 x (20 x 5.5) and that of the base = (20 x 10 ) and so the total area to be pained = 2 x (10 x 5.5) + 2 x (20 x 5.5) + (20 x 10 ) = 530cm3 Since, the total expenses of painting this area is ₹70, the rate of painting = 70/530 = 0.13 = ₹ 0.1 per cm3 (approx)

Question 2

A wooden box of dimensions 8 m x 7 m x 6 m is to carry rectangular boxes of dimensions 8 cm x 7 cm x 6 cm. The maximum number of boxes that can be carried in 1 wooden box is?

Accepted Answer

Number of boxes = (Volumes of wooden box in cm3) / (Volume of 1 small box) = (800 x 700 x 600) / (8 x 7 x 6) = 1000000

Question 3

If the length breadth and the height of cuboid are in the ratio 6 :5 : 4 and if the total surface area is 33300 cm 2, then the length, breadth and height in cm. are respectively?

Accepted Answer

Let length = 6k, breadth = 5k and height = 4k in cm ∵ 2(6k x 5k + 5k x 4k + 6k x 4k) = 33300 ⇒ 148k2 = 33300 ⇒ k2 = 33300/148 = 225 ⇒ k = 15 ∴ Length = 90 cm, Breadth = 75 cm, Height = 60 cm

Question 4

If the length of diagonal of a cube is 8√ 3 cm, then its surface area is?

Accepted Answer

∵ √3a = 8√3 ⇒ a = 8 ∴ Surface area = 6a2 = (6 x 8 x 8) cm2 = 384 cm2

Question 5

The length of longer pole that can be placed on the floor of a room is 10 m and the length of the longest pole that can be placed in the room is 10√ 2m. The height of the room is?

Accepted Answer

∵ l2 + b2 = (10)2 = 100 And l2 + b2 + h2 = (10√300)2 = 200 ⇒ h2 = (200 - 100) = 100 ∴ h = 10 m

Question 6

The length of diagonal of a cube is (14 x √ 3) cm. The volume of the cube is?

Accepted Answer

∵ √3 a = 14 x √3 ⇒ a = 14 ∴ Volume of the cube = (14 x 14 x 14)cm3 = 2744 cm3

Question 7

The maximum length of a pencil that can be kept in a rectangular box of dimensions 8 cm x 6 cm x 2 cm is?

Accepted Answer

Length of pencil = √ [(8)2 + (6)2 + (2)2] cm = √104 cm = 2√26

Question 8

A river 2 m deep and 45 m wide is running at the rate of 3 km/hr. The amount of water that runs into the per minute is?

Accepted Answer

Speed per min = (3 x 1000)/60 m = 50 m Volume of water running per min. = (45 x 2 x 50) m2 = 4500 m3

Question 9

Bricks are worth Rs. 750 per 1000 and their length, breadth and height 25 cm, 12.5 cm and 7.5 cm respectively. The cost of bricks required to build a wall 200 m long, 1.8 m high and 37.5 cm thick is?

Accepted Answer

Number of bricks required = Volume of wall in cm3 / Volume of 1 brick = (200 x 100 x 1.8 x 100 x 37.5)/(25 x 12.5 x 7.5) = 57, 600 ∴ Required cost = Rs. (750 x 57600)/1000 = Rs. 43, 200

Question 10

A metal sheet 27 cm long, 8 cm broad and 1 cm thick is melted into a cube. The difference between the surface area of two solids is?

Accepted Answer

Volume of cube formed = 216 cm3 ∴ Edge of the cube = (6 x 6 x 6)1/3 = 6 cm Surface area of original metal sheet = 2(27 x 8 + 8 x 1 + 27 x 1) cm2 = 502 cm2 Surface area of the cube formed = [6 x (6)2] cm2 = 216 cm2 ∴ Required difference in area of two solids = (502 - 216) cm2 = 286 cm2

Question 11

The radius of a wire is decreased to one-third. If volume remains the same, length will increase?

Accepted Answer

Let original radius = r and original length = h New radius = r/3 Let new length = H Then, π r2h = π (r/3)2 x H = π r2H/9 ∴ H = 9h Thus, the length becomes 9 times .

Question 12

The Volume of a wall, 5 times as high as it is broad and 8 times as long as it is high, is 12.8 m 3. The breadth of the wall is?

Accepted Answer

Let breadth = L meters. Then, height = 5L meters and length = 40L meters ∴ L x 5L x 40L = 12.8 ⇒ L3 = 12.8/200 = 128/2000 = 64/1000 ∴ L = 4/10 Thus, breadth = (4/10) m = (4 x 100)/10 cm = 40 cm

Question 13

The volume of a sphere is 38808 cu. cm. The curved surface area of the sphere is?

Accepted Answer

∵ 4/3 x 22/7 x r3 = 38808 ⇒ r3 = (38808 x 7/22 x 3/4) = (21)3 ∴ r = 21 cm So, curved surface area of sphere = 4πr2 = (4 x 22/7 x 21x21) cm2 = 5544 cm2

Question 14

A right cylindrical vessel is full with water. How many right cones having the same diameter and height as those of the right cylinder will be needed to store that water?

Accepted Answer

Let N cones be needed Then, 1/3 πr2 h x N = πr2h ⇒ N = 3

Question 15

A right 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?

Accepted Answer

Let the height of cylinder = h and height cone = H Then, πr2 h = 1/3 πr2 H ⇒ h/H = 1/3 = 1 : 3

Question 16

A cylindrical vessel of radius 4 cm contains water. A solid sphere of radius 3 cm is lowered into the water until it is completely immersed. The water level in the vessel will rise by?

Accepted Answer

∵ π x (4)2 x h = 4/3 π x (3)3 ∴ h = 9/4 cm = 2.25 cm

Question 17

The radius of a circular cylinder is the same as that of a sphere. Their volumes are equal. The height of the cylinder is?

Accepted Answer

∵ 4/3 πr3 = πr2 h ⇒ h = 4/3 r ∴ Height = 4/3 times its radius.

Question 18

A hollow garden roller 63 cm wide with a girth of 440 cm is made of iron 4 cm thick. The volume of iron used is?

Accepted Answer

∵ Circumference of the girth = 440 cm ⇒ 2πr = 440 ⇒ r = 440 / (2 x 7/22) = 70 cm Thus, Outer radius = 70 cm Inner radius = (70 - 4) cm = 66 cm Volume of iron = π[(70)2 - (66)2] x 63 cm3 = (22/7) x 136 x 4 x 63 cm3 = 107712 cm 3

Question 19

The number of solid spheres each of diameter 6 cm that could be moulded to form a solid metal cylinder 4 cm, is?

Accepted Answer

Let the number of spheres be N Then, N x 4/3 π x (3)3 = π x (2)2 x 45 ⇒ 36N = 180 ∴ N = 180/36 = 5

Question 20

The length of the wire of 0.2 mm radius that can be drawn after melting a solid copper sphere of diameter 18 cm, is?

Accepted Answer

Radius of sphere = 9 cm Volume of sphere = [ 4/3 x π x (9)3] cm3 = 972π cm3 Radius of wire = 0.2 mm = 2/(10 x 10) cm = 1/50 cm Let the length of wire be = L cm Then, 972π = π x (1/50)2 x L ⇒ L = (972 x 50 x 50 ) cm ∴972π = π x (1/50)2 x L ⇒ L = (972 x 50 x 50) cm ∴ Length of wire = (972 x 50 x 50)/100 m = 24, 300 m

Volume and Surface Area Questions