Volume & Surface Area of Solids — Case-Study Based Questions

Case Study
The length, breadth and height of Kavita's bedroom are 6 m, 4 m and 3 m respectively. It has two equal windows, each of dimensions 1 m × 0.5 m. It also has a door of dimensions 2 m × 1 m.

Based on the above information, answer the following questions:

1. Area occupied by the door and the two windows is :
   (a) 1 m²
   (b) 2 m²
   (c) 2.5 m²
   (d) 3 m²

2. Kavita wants to whitewash the four walls of the room. Area to be whitewashed is :
   (a) 60 m²
   (b) 57 m²
   (c) 55 m²
   (d) 50 m²

3. Square tiles each of side 50 cm are laid on the floor of the room. The number of such tiles laid is :
   (a) 100
   (b) 98
   (c) 96
   (d) 72

4. Volume of air contained in the room is :
   (a) 72 m³
   (b) 70 m³
   (c) 60 m³
   (d) 52 m³

5. The length of the longest rod (to the nearest m) that can be placed in the room is :
   (a) 4 m
   (b) 5 m
   (c) 8 m
   (d) 7 m

Answer

Given,

Length (l) = 6 m

Breadth (b) = 4 m

Height (h) = 3 m

Dimension of each window = 1 m × 0.5 m

Dimension of door = 2 m × 1 m

1. Area of one window = 1 × 0.5 = 0.5 m²

∴ Area of two windows = 2 (1 × 0.5) = 1 m²

Area of one door = 2 × 1 = 2 m².

Total area = 2 + 1 = 3 m².

Hence, option (d) is the correct option.

2. Area to be whitewashed = Area of four walls - Area of two windows and a door.

Calculating the area of four walls,

We know that,

Area of four walls = 2h(l + b)

= 2 × 3 × (6 + 4)

= 6 × 10

= 60 m².

Area of two windows and a door = 3 m².

Area to be whitewashed = 60 - 3 = 57 m².

Hence, option (b) is the correct option.

3. Given,

Tile side = 50 cm = 0.5 m

Calculating the floor area,

Area of floor = Length × Breadth

= 6 × 4

= 24 m².

Area of square tile = (side)²

= (0.5)²

= 0.25 m²

Number of tiles = $\dfrac{\text{Area of floor}}{\text{Area of each tile}}$

= $\dfrac{24}{0.25}$

= 96.

Hence, option (c) is the correct option.

4. Volume of air contained in the room = l × b × h

= 6 × 4 × 3

= 72 m³.

Hence, option (a) is the correct option.

5. Length of the longest rod that can be placed in the room = Diagonal of room

$\text{Diagonal (d)} = \sqrt{l^2 + b^2 + h^2} \\[1em] = \sqrt{6^2 + 4^2 + 3^2} \\[1em] = \sqrt{36 + 16 + 9} \\[1em] = \sqrt{61} \\[1em] = 7.81 \approx 8 \text{ m}.$

Hence, option (c) is the correct option.

Case Study
Manish is a carpenter. One day, he made an open cubical box of internal edge 18 cm. The thickness of the plywood is 1 cm. He painted the inner surface of the box black and the outer lateral surfaces as green.

Based on the above information, answer the following questions:

1. Total area to be painted black is :
   (a) 1600 cm²
   (b) 1620 cm²
   (c) 1650 cm²
   (d) 1680 cm²

2. Length of the outer edge of the box is :
   (a) 20 cm
   (b) 19 cm
   (c) 18 cm
   (d) 18.5 cm

3. Total outer lateral surface area to be painted green is :
   (a) 1500 cm²
   (b) 1510 cm²
   (c) 1520 cm²
   (d) 1600 cm²

4. Total outer surface area of the box (excluding the top) is :
   (a) 1900 cm²
   (b) 1920 cm²
   (c) 2000 cm²
   (d) 2320 cm²

5. Length of the longest rod that can fit inside the box is :
   (a) 18 cm
   (b) 19 cm
   (c) 20 cm
   (d) 3 cm

Answer

Given,

Internal edge of cube = 18 cm

Thickness of plywood = 1 cm

1. Inside the open cubical box there are 5 faces (4 sides + 1 bottom)

Calculating area of one face,

Area = (side)²

= (18)²

= 324 cm².

Calculating total inner area,

Total inner area = 5 × 324 = 1620 cm².

Thus, the total area to be painted black is 1620 cm².

Hence, option (b) is the correct option.

2. Thickness 1 cm will be on both sides of the box.

∴ Outer edge = Internal edge of cube + 2(Thickness)

= 18 + 2(1)

= 18 + 2 = 20 cm.

Hence, option (a) is the correct option.

3. Outer edge = 20 cm

Lateral surface area refers to the 4 side walls only (not the bottom).

Outer lateral area = 4 × (outer edge)²

= 4 × (20)²

= 4 × 400

= 1600 cm².

Hence, option (d) is the correct option.

4. Total outer surface area of the box (excluding the top)

Total faces excluding the top = 5

Area of one face = (20)² = 400 cm².

Calculating the area of all the faces,

Area of all the faces = 5 × 400 = 2000 cm².

Hence, option (c) is the correct option.

5. The rod must lie completely inside the open box, so the maximum straight length inside the box equals the internal edge length = 18 cm.

Hence, option (a) is the correct option.