Volume & Surface Area of Solids — Multiple Choice Questions

If the length of each side of a cube is reduced by 25%, then the ratio of the volumes of the original and the new cube is :

1. 64 : 1

2. 4 : 3

3. 64 : 27

4. 32 : 9

Answer

Let the original side of cube be a units and side of cube after reduction be a'.

According to the question,

a' = a - 25% of a = a - 0.25a = 0.75a = $\dfrac{3}{4}a$.

We know that,

Volume of cube = (side)³.

Calculating original volume,

V₁ = a³.

Calculating the volume after reduction,

V₂ = (a')³

= $\Big(\dfrac{3}{4}a\Big)$³

= $\dfrac{27}{64}$ a³

Ratio of original volume to the new volume

V₁ : V₂

a³ : $\dfrac{27}{64}$ a³

$\dfrac{a^3}{\dfrac{27}{64}a^3}$

$\dfrac{64}{27}$

64 : 27.

Hence, option 3 is the correct option.

If the length of each side of a cube is reduced by 50%, then the ratio of the total surface area of the original and the new cube is :

1. 2 : 1

2. 4 : 1

3. 8 : 1

4. 8 : 3

Answer

Let the original side be a units and new side be a' units.

Reduction of 50% :

a' = a - 50% of a = a - $\dfrac{1}{2}a = \dfrac{a}{2}$ units.

We know that,

Total surface area of cube (TSA) = 6(side)².

Calculating the original total surface area of cube,

original TSA = 6a²

Calculating the new total surface area of a cube,

new TSA = 6(a')²

= 6 $\Big(\dfrac{a}{2}\Big)^2$

= 6 × $\dfrac{a^2}{4}$

= $\dfrac{3a^2}{2}$.

Ratio of TSA of original cube to the new cube:

original TSA : new TSA

6a² : $\dfrac{3a^2}{2}$

$\dfrac{6a^2}{\dfrac{3a^2}{2}}$

$\dfrac{4}{1}$

4 : 1.

Hence, option 2 is the correct option.

The length and the breadth of a cuboid are 60 cm and 50 cm respectively. If the total surface area of the cuboid is 14800 cm², then its height is :

1. 40 cm

2. 32 cm

3. 25 cm

4. 30 cm

Answer

Given,

Length = 60 cm

Breadth = 50 cm

Total surface area = 14800 cm².

By formula,

Total surface area of cuboid = 2(lb + bh + hl)

⇒ 14800 = 2(60 × 50 + 50 × h + h × 60)

⇒ 14800 = 2(3000 + 50h + 60h)

⇒ 3000 + 50h + 60h = $\dfrac{14800}{2}$

⇒ 3000 + 110h = 7400

⇒ 110h = 7400 - 3000

⇒ 110h = 4400

⇒ h = $\dfrac{4400}{110}$ = 40 cm.

Hence, option 1 is the correct option.

The base of a cuboid is a square and its height is 4 cm. If the volume of the cuboid is 200 cm³, then the length of the base is :

1. $4\sqrt{2}$ cm

2. 8 cm

3. 5 cm

4. $5\sqrt{2}$ cm

Answer

The base is a square having length and breadth both equal to x cm.

Given,

Height = 4 cm

Volume = 200 cm³.

We know that,

Volume = length × breadth × height

⇒ 200 = x × x × 4

⇒ 200 = x² × 4

⇒ x² = $\dfrac{200}{4}$

⇒ x² = 50

⇒ x = $\sqrt{50}$

⇒ x = $\sqrt{25 × 2}$

⇒ x = $5\sqrt{2}$ cm.

Hence, option 4 is the correct option.

The length of the diagonal of a cube is $16\sqrt{3} \text{ m}$. The volume of the cube is :

1. 4096 m³

2. 4196 m³

3. 3146 m³

4. 4036 m³

Answer

Given,

Diagonal of cube (d) = $16\sqrt{3} m$.

Let side of cube be a meters.

Diagonal (d) = $a\sqrt{3}$

⇒ $16\sqrt{3} = a\sqrt{3}$

⇒ a = 16 m.

Calculating the volume of cube,

Volume of cube = a³

= 16³

= 4096 m³.

Hence, option 1 is the correct option.

The total surface area of a cube is 96 cm². The length of a diagonal of the cube is :

1. 16 cm

2. $8\sqrt{3}$ cm

3. $6\sqrt{3}$ cm

4. $4\sqrt{3}$ cm

Answer

Given,

Total surface area of cube = 96 cm².

By formula,

Total surface of a cube = 6a²

⇒ 96 = 6a²

⇒ a² = $\dfrac{96}{6}$

⇒ a² = 16

⇒ a = $\sqrt{16}$

⇒ a = 4 cm.

Calculating the length of diagonal of a cube,

Diagonal of a cube = $a\sqrt{3} = 4\sqrt{3}$ cm.

Hence, option 4 is the correct option.

The length of the diagonal of a cube is $6\sqrt{6} \text{ cm}$. The total surface area of the cube is :

1. 216 cm²

2. 432 cm²

3. 1296 cm²

4. 1548 cm²

Answer

Given,

Length of the diagonal of a cube = $6\sqrt{6}$ cm

Calculating the side of a cube,

Diagonal of a cube = $a\sqrt{3}$

⇒ $6\sqrt{6} = a\sqrt{3}$

⇒ a = $\dfrac{6\sqrt{6}}{\sqrt{3}}$

⇒ a = $6\sqrt{2}$ cm.

Calculating the total surface area of a cube,

Total surface area of a cube = 6a²

= 6 × $(6\sqrt{2})^2$

= 6 × 36 × 2

= 432 cm².

Hence, option 2 is the correct option.

The volume of a cube is 125 m³. The total surface area of the cube is :

1. 150 m²

2. 250 m²

3. 375 m²

4. 432 m²

Answer

Given,

Volume of cube = 125 m³

Calculating the side of a cube,

Volume of cube = a³

⇒ 125 = a³

⇒ a = $\sqrt[3]{125}$

⇒ a = 5 m.

Calculating the total surface area of the cube,

Total surface area of the cube = 6a²

= 6 × 5²

= 6 × 25

= 150 m².

Hence, option 1 is the correct option.

460 cm² of metal sheet is needed to make a closed box of length 12 cm and height 5 cm. The breadth of the box is :

1. 10 cm

2. 12 cm

3. 14 cm

4. 16 cm

Answer

Given,

Length(l) = 12 cm

Height(h) = 5 cm

Total surface area of sheet required to make a closed box = 460 cm²

Let breadth of the box be b cm.

Calculating the breadth of the box,

Total surface area = 2(lb + bh + hl)

⇒ 460 = 2(12 × b + b × 5 + 5 × 12)

⇒ 460 = 2(12b + 5b + 60)

⇒ 460 = 2(17b + 60)

⇒ 17b + 60 = $\dfrac{460}{2}$

⇒ 17b + 60 = 230

⇒ 17b = 230 - 60

⇒ 17b = 170

⇒ b = $\dfrac{170}{17}$ = 10 cm.

Hence, option 1 is the correct option.

The length of the largest rod that can be kept in a room of length 5 m, breadth 4 m and height 3 m is :

1. $3\sqrt{2}\text{ m}$

2. $5\sqrt{2}\text{ m}$

3. $7\sqrt{2}\text{ m}$

4. $\sqrt{2}\text{ m}$

Answer

Given,

Length (l) = 5 m

Breadth (b) = 4 m

Height (h) = 3 m

Length of the largest rod that can fit in the room is the diagonal of the room.

Calculating the length of the diagonal of the room (cuboid),

$\text{Diagonal of room (d)} = \sqrt{l^2 + b^2 + h^2} \\[1em] = \sqrt{5^2 + 4^2 + 3^2} \\[1em] = \sqrt{25 + 16 + 9} \\[1em] = \sqrt{50} \\[1em] = \sqrt{25 × 2} \\[1em] = 5 \sqrt{2} \text{ m}.$

Hence, option 2 is the correct option.

The sum of the length, breadth and height of a cuboid is 24 cm, and the length of its diagonal is 15 cm. The area of its total surface is :

1. 348 cm²

2. 349 cm²

3. 350 cm²

4. 351 cm²

Answer

Given,

Sum of dimensions : l + b + h = 24

Length of diagonal (d) = 15 cm.

We know that,

Diagonal of cuboid (d) = $\sqrt{l^2 + b^2 + h^2}$

⇒ 15 = $\sqrt{l^2 + b^2 + h^2}$

Squaring on both sides,

⇒ 15² = $(\sqrt{l^2 + b^2 + h^2})^2$

⇒ 225 = l² + b² + h²

By formula,

(l + b + h)² = l² + b² + h² + 2(lb + bh + hl)

By substituting the values we get,

⇒ (24)² = 225 + 2(lb + bh + hl)

⇒ 576 = 225 + 2(lb + bh + hl)

⇒ 2(lb + bh + hl) = 576 - 225

⇒ 2(lb + bh + hl) = 351

Since, Total surface area of cuboid = 2(lb + bh + hl)

∴ Total surface area of cuboid = 351 cm².

Hence, option 4 is the correct option.

The volume of a cuboid whose length, breadth and height are 8 cm, 5 cm and 3 cm respectively is :

1. 120 cm³

2. 122 cm³

3. 124 cm³

4. 128 cm³

Answer

Given,

Length (l) = 8 cm

Breadth (b) = 5 cm

Height (h) = 3 cm

Calculating the volume of cuboid,

Volume of cuboid = l × b × h

= 8 × 5 × 3

= 120 cm³.

Hence, option 1 is the correct option.

The area of cross-section of a hosepipe is 3 cm². Water flows through it at a speed of 50 cm/sec. How many litres of water flows out of it in one minute?

1. 7 litres

2. 8 litres

3. 9 litres

4. 11 litres

Answer

Given,

Area of cross-section = 3 cm².

Speed of water = 50 cm/sec.

1 minute = 60 seconds

Calculating the distance traveled by water in 1 minute,

Distance = Speed × Time

Distance = 50 cm/sec × 60 sec

= 3000 cm.

Calculating the volume of water,

Volume = Area of cross-section × Distance by water in 1 minute

= 3 × 3000

= 9000 cm³.

1000 cm³ = 1 litre

∴ 9000 cm³ = $\dfrac{9000}{1000}$ litres

= 9 litres.

Hence, option 3 is the correct option.

The sum of the length, breadth and height of a cuboid is 41 cm. If the length of its diagonal is 25 cm, then its total surface area is :

1. 1050 cm²

2. 1052 cm²

3. 1054 cm²

4. 1056 cm²

Answer

Given,

Sum of sides : l + b + h = 41 cm.

Length of diagonal (d) = 25 cm.

We know that,

Diagonal of cuboid (d) = $\sqrt{l^2 + b^2 + h^2}$

⇒ 25 = $\sqrt{l^2 + b^2 + h^2}$

Squaring on both sides,

⇒ 25² = $(\sqrt{l^2 + b^2 + h^2})^2$

⇒ 625 = l² + b² + h²

By formula,

(l + b + h)² = l² + b² + h² + 2(lb + bh + hl)

By substituting the values we get,

⇒ (41)² = 625 + 2(lb + bh + hl)

⇒ 1681 = 625 + 2(lb + bh + hl)

⇒ 2(lb + bh + hl) = 1681 - 625

⇒ 2(lb + bh + hl) = 1056

Since, Total surface area of cuboid = 2(lb + bh + hl)

∴ Total surface area of cuboid = 1056 cm².

Hence, option 4 is the correct option.

The weight of a rectangular box with lid is 60 kg. The box filled with water weighs 600 kg. The weight of 1 litre of water is 1.2 kg. If the thickness of the box is 5 cm, and the external length and breadth of the box are 16 dm and 8.5 dm respectively, then the external height of the box is :

1. 5 dm

2. 6 dm

3. 7 dm

4. 8 dm

Answer

Given,

External length of rectangular box = 16 dm

External breadth of rectangular box = 8.5 dm

Thickness = 5 cm = 0.5 dm

Density of water = 1.2 kg/litre

Weight of box with lid = 60 kg

Weight of box filled with water = 600 kg

Calculating the weight of the water,

Weight of the water = Weight of filled box - Weight of empty box.

= 600 - 60 = 540 kg.

Calculating the volume of water,

Volume of water = $\dfrac{\text{Weight of the water}}{\text{Density of the water}}$

= $\dfrac{540}{1.2}$

= 450 litres.

1 litre = 1 dm³

∴ 450 litres = 450 dm³.

Calculating internal dimensions,

Internal length = External length - 2 × Thickness

= 16 - (2 × 0.5)

= 16 - 1 = 15 dm.

Internal breadth = External breadth - 2 × Thickness

= 8.5 - (2 × 0.5)

= 8.5 - 1 = 7.5 dm.

Calculating the internal volume of rectangular box,

Volume of cuboid = l × b × h

⇒ 450 = 15 × 7.5 × h

⇒ 450 = 112.5 × h

⇒ h = $\dfrac{450}{112.5}$

⇒ h = 4

∴ Internal height = 4 dm.

Calculating the external height of the box,

Since the box has thickness at both the top and bottom,

So,

External height = Internal height + 2(Thickness)

= 4 + 2(0.5)

= 4 + 1

= 5 dm.

Hence, option 1 is the correct option.