Question 1

The cost of painting the four walls of a room is Rs. 350. The cost of painting a room three times in length, breadth and height respectively will be?

Accepted Answer

Area of 4 walls of the room = [2 (l + b ) x h ]m2 Area of 4 walls of new room = [2 (3l + 3b) x 3h]m2 = 9 x [2(l + b) x h ]m2 ∴ Cost of painting the 4 walls of the new room = Rs. (9 x 350) = Rs. 3150

Question 2

A solid consists of a circular cylinder with an exact fitting right circular cone placed on the top. The height of the cone is h. If the total volume of the cone, then the height of the cylinder is?

Accepted Answer

Let the height of the cylinder be H and its radius = r Then, (πr2 H) + (1/3) x (πr2h) = 3 x (1/3)πr2h ⇒ πr2H = (2/3)πr2h ∴ H = 2h/3.

Question 3

The diagonal of a cube is 2√ 3 cm. Find the surface area of the cube.

Accepted Answer

Given, diagonal = 2√3 ⇒ a√3 = 2√3 [a = edge of the cube] ∴ a = 2 ∴ Required surface area = 6a2 = 6 x 22 = 24 sq cm

Question 4

If the total surface area of a cube is 6 sq units, then what is the volume of the cube?

Accepted Answer

Total surface area of a cube = 6a2 ⇒ 6 = 6a2 ⇒ a2 = 1 ∴ a = 1 Now, volume of the cube = a3 = 13 = 1 cu unit

Question 5

If the volume of a cube is 729 cu cm, what is the length of its diagonal?

Accepted Answer

Volume of cube = (Side)3 ∴ 729 = a3 ⇒ a = 9 cm Diagonal of cube = side x √3 = 9 x √3 = 9√3 cm

Question 6

Find the surface area of a cuboid 10 m long, 5 m broad and 3 m high?

Accepted Answer

Given that, l = 10 m , b = 5 m, h = 3 m lb = 10 x 5 = 50, bh = 5 x 3 = 15, lh = 10 x 3 = 30 ∴ Surface area of a cuboid = 2(lb + bh + lh) = 2(50 + 15 + 30) = 2 x 95 = 190 sq m

Question 7

The surface area of a cube is 726 sq cm. Find the volume of the cube.?

Accepted Answer

According to the question, 6a2 = 726 [a = edge of the cube] ⇒ a2 = 726/6 = 121 ∴ a = √121 = 11 cm ∴ Required volume = a3 = 113 = 1331 cm3

Question 8

The volume of a cuboid is 1989 cm 3 , while its length and breadth are 17 cm and 13 cm, respectively, Find the height of the cuboid.?

Accepted Answer

Volume = 1989 ⇒ lbh = 1989 ⇒ 17 x 13 x h = 1989 ∴ h = 1989/(17 x 17) = 9 cm

Question 9

Three cubes of sides 1 cm, 6 cm and 8 cm are melted to form a new cube. Find half of the surface area of the new cube.?

Accepted Answer

Volume (new cube) = (13 + 63 + 83 ) = 729 cm3 a3 = 729 ⇒ a = ∛ 729 ∴ Surface area of the new cube = 6a2 = 6 x 92 = 486 sq cm Surface area/2 = 486/2 = 243 sq cm

Question 10

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 largest pencil that can be kept in a box = Diagonal of box = √l2 + b2 + h2 where, l = 8 cm, b = 6 cm , h = 2 cm = √64 + 36 + 4 = √104 = 2√26

Question 11

Internal length, breadth and height of a rectangular box are 10 cm, 8 cm and 6 cm, respectively. How many boxes are needed which can be packed in a cube whose volume is 6240 cu. cm.?

Accepted Answer

Volume of rectangular box = 10 x 8 x 6 = 480 cm3 Volume of cubes = 6240 cm3 ∴ Required boxes = Volume of cubes / Volume of rectangular box = 6240/480 = 13 Hence, 13 boxes are needed.

Question 12

A metal box measures 20 cm x 12 cm x 5 cm. Thickness of the metal is 1 cm. Find the volume of the metal required to make the box.?

Accepted Answer

External volume = 20 x 12 x 5 = 1200 cm3 Internal volume = (20 - 2) x (12 - 2) x (5 - 2) = 18 x 10 x 3 = 540 cm3 ∴ Volume of the metal = External volume - Internal volume = 1200 - 540 = 660 cm3

Question 13

How many cubes of 3 cm edge be cut out of a cube of 18 cm edge?

Accepted Answer

Let the number of cubes that can be cut from bigger cube = n. Then, n x Volume of smaller cube = Volume of bigger cube Edge of smaller cube = 3 cm Edge of bigger cube = 18 cm n x (3)3 = (18)3 ⇒ n = (18 x 18 x 18) / (3 x 3 x 3) ∴ n = 6 x 6 x 6 = 216 cubes

Question 14

A cube has each edge 2 cm and a cuboid is 1 cm long, 2 cm wide and 3 cm high. The paint in a certain container is sufficient to paint an area equal to 54 cm 2. Which one of the following is correct?

Accepted Answer

Surface area of cube which can be painted = 6(Side)2 = 6(2)2 = 24 cm2 Now, surface area of cuboid which can be painted = 2(lb + bh + lh) = 2(2 + 6 + 3) = 22 cm Total surface area of both cube and cuboid = 22 + 24 = 46 cm2 < 54 cm2 Therefore, both cube and cuboid can be painted.

Question 15

The whole surface area of a rectangular block is 8788 sq cm. If length, breadth and height are in ratio of 4 : 3 : 2, then find the length.?

Accepted Answer

Let length, breadth and height be 4k, 3k and 2k, respectively. Whole surface area = 2(lb + bh + lh) ⇒ (lb + bh + lh) = 8788/2 = 4394 ⇒ (4 x 3 + 3 x 2 + 2 x 4 ) k2 = 4394 ⇒ 26k2 = 4394 ⇒ k2 = 169 ⇒ k = 13 ∴ Length = 4k = 4 x 13 = 52 cm

Question 16

If the area of three advancement faces of a cuboidical box 120 sq cm, 72 sq cm and 60 sq cm, respectively, then find the volume of the box.?

Accepted Answer

Given, lb = 120 sq cm., bh = 72 sq cm, lh = 60 sq cm. ∴ lb x bh x lh = 120 x 72 x 60 ⇒ (lbh)2 = 120 x 72 x 60 ⇒ lbh = √120 x 72 x 60 = √12 x 10 x 12 x 6 x 10 x 6 = 12 x 10 x 6 = 720 cm3

Question 17

The capacity of a cuboid tank of water is 50000 L. Find the breadth of the tank, if its length and depth are 2.5 m and 10 m, respectively.

Accepted Answer

Capacity of tank = = 50000 L = 50 m3 [∴ 1L = 1/1000 m3 ] ∴ Breadth = 50/(2.5 x 10) = 2 m

Question 18

The paint in certain container is sufficient to paint an area equal to dimensions 22.5 cm x 10 cm x 7.5 cm can be painted out of this container?

Accepted Answer

Surface area of 1 brick = 2 (lb + bh + lh) = 2(22.5 x 10 + 10 x 7.5 + 7.5 x 22.5) = 2(225 + 75 + 168.75) = 2 x 468.75 = 937.50 cm2 = 93750/(100 x 100) = 0.09375 sq m ∴ Number of bricks = Total area/Surface area of 1 brick = 9.375 / 0.09375 = 100

Question 19

What are the dimensions (length, breadth and height, respectively) of a cuboid with volume 720 cu cm, surface area 484 sq cm and the area of the base 72 sq cm?

Accepted Answer

Volume of the cuboid = 720 cm3 Height of the cuboid = Volume of the cuboid / Base area of the cuboid = 720 / 72 = 10 cm [By Hit Trail] Surface area of the cuboid = 2 (lb + bh + hl) = 2(9 x 8 + 8 x 10 + 10 x 9 ) = 2 (72 + 80 + 90) = 2 x 242 = 484 cm2 ∴ It is obvious that length, breadth and height of the cuboid is 9 cm, 8 cm and 10 cm.

Question 20

The volume of a cube is numerically equal to sum of its edges. What is the total surface area in square units?

Accepted Answer

Let the edge of a square be x. and sum of its edges = 12 x Now, by condition, x3 = 12x ⇒ x(x2 - 12) = 0 ⇒ Its total surface area = 6x2 = 6(12) = 72 sq units

Volume and Surface Area Questions