Numerical Ability Test - 15

Concept:

• Diagonal of cube = a√3

• Surface area of cube = 6a2 

Calculation:

We know that, 

Diagonal of cube = a√3

⇒ a√3 = 8√3 

⇒ a = 8 

Now, surface area of cube = 6a² = 6(8)² = 384 cm².

Hence, the surface area of the cube is 384 cm2.

Concept:

Volume of cuboid = length × breadth × height

Calculation:

Let the edges of the cuboid be a, b, and c.

Since the areas of three adjacent faces are in the ratio 7 : 10 : 14,

⇒ ab : bc : ca = 7 : 10 : 14

∴ ab : bc = 7 : 10

⇒ a : c = 7 : 10      

Also, bc : ca = 10 : 14

⇒ b : a = 10 : 14 = 5 : 7   

⇒ a : b = 7 : 5      

∴ a : b : c = 7 : 5 : 10

Let the ratios be 7x, 5x and 10x.

⇒ Volume of cuboid = 350 cm³ 

⇒ abc = 350

⇒ (7x) × (5x) × (10x) = 350

⇒ x³ = 1

⇒ x = 1

So, the edges are 7 cm, 5 cm and 10 cm.

Hence, the longest side has the length 10 cm.

Concept Used:

To find the area of  visible faces, we have to take the area of 2 sides

Area of rectangle = Length × Breadth

Calculation:

Area of ABJI = (15 × 7) cm²

⇒ 105 cm²

Area of IJGH = (7 × 5) cm²

⇒ 35 cm²

Area of BCLK = (13 × 8) cm²

⇒ 104 cm²

Area of KLEF = (13 × 5) cm²

⇒ 65 cm²

Area of KFGJ = (5 × 7) cm²

⇒ 35 cm²

Area of CDEL = (8 × 5) cm²

⇒ 40 cm²

Total area of visible faces = (105 + 35 + 104 + 65 + 35 + 40) cm²

⇒ 384 cm²

∴ Total area of visible faces is 384 cm²

Given:

Dimensions of a cuboid are, l × b × h = 27 cm × 18 cm × 21 cm 

Formulas used:

Volume of cuboid = l × b × h 

Volume of cube with side a = a³

Calculation:

No of cube to be cut = Volume of cuboid/Volume of cube

⇒ (27 cm × 18 cm × 21 cm)/(3 × 3 × 3)

⇒ 18 × 21 

⇒ 378

∴ The no of cubes that can be cut out of cuboid = 378

Given:

Volume of the Reservoir = 84000 L = 84 m³

Length = 5 m and breadth = 2.1 m (can take vice versa also)

Formula used:

The volume of Cuboid = Length × Breadth × Height

Calculation:

Let 'h' be the depth of the reservoir (Depth can be considered as height)

84 = 5 × 2.1 × h

⇒ h = 84/(5 × 2.1)

⇒ h = 8 m

∴ The depth of water in the reservoir is 8 m

Given:

The height of the cylinder = 21 cm

The radius of the cylinder = 5 cm

Formula used:

The curved surface area of the cylinder = 2π r h

Here, r is the radius of the cylinder and h is the height of the cylinder

Calculation:

The curved surface area of the cylinder = 2π r h

⇒ 2 × (22/7) × 5 × 21

⇒ 2 × 22 × 5 × 3

⇒ 660 cm²

Given:

Radius = 8 cm

Radius of small spheres = 0.5 cm

Concept used:

The volume of original sphere = The total volume of all small spheres

Formula used:

Total volume of sphere = (4/3)πr3

where, r = radius of sphere

Calculations:

Let the number of small sphere be n.

The volume of sphere melted = (4/3) × (22/7) × (8)³ 

The volume of all small spheres = (4/3) × (22/7) × (0.5)3

⇒ (4/3) × (22/7) × (0.5)³ × n = (4/3) × (22/7) × (8)3

⇒ (1/2)³ × n = 512

⇒ n = 512 × 8

⇒ n = 4096

∴ The number of small spheres made is 4096.

Given:

Capacity of tank = 6160 m³

Diameter of tank = 28 m

Formula used:

Volume of cylinder = πr²h

Calculation:

Radius of tank = 28/2 m 

⇒ 14 m

Volume of tank = πr²h

⇒ 22/7 × (14)² × h = 6160

⇒ 22/7 × 196 × h = 6160

⇒ 22 × 28 × h = 6160

⇒ 616 × h = 6160

⇒ h = 10 m

∴ The depth of the tank is 10

Given:

Surface area of sphere = 1,386 cm²

Concept used:

Surface area of sphere = 4πr²

π = 22/7

Calculation:

Surface area of sphere = 4πr2

⇒ 1,386 cm² = 4 × (22/7) × r² 

⇒ (1,386 × 7)/(4 × 22) = r²

⇒ (63 × 7)/4 = r²

⇒ 441/4 = r²

⇒ (21/2)² = r²

⇒ 21/2 = r

⇒ 10.5 cm = r

∴ The radius of the sphere is 10.5 cm.

Important Points

Volume of sphere = (4/3)πr³ Cubic units

Volume of hollow sphere = [(4/3)π(R³ - r³)] cubic units 

where, r = Inner radius and R = Outer radius

Given:

The volume of a cylinder is 4312 cm3

Curved surface area = Total surface area/3

Formula Used:

Volume of cylinder = πr²h

Curved Suface Area = 2πrh 

Total Suface Area = 2πr(h + r)

Calculation:

According to the question 

Curved surface area = Total surface area/3

⇒ 2πrh = 2πr(h + r)/3

⇒ 3h = (h + r)

⇒ 2h = r     -----(i)

Now, we have 

The volume of a cylinder = 4312 cm3

⇒ πr2h = 4312

⇒ (22/7) r2 (r/2) = 4312

⇒ r³ = 14³

⇒ r = 14

So, h = r/2 = 14/2 = 7 cm

Now, Curved Suface Area = 2πrh

⇒ 2 × (22/7) × 14 × 7

⇒ 616 cm²

∴ The curved surface area of cylinder is 616 cm².