Numerical Ability Test - 29

Given:

Length is decreased by 10%.

Breadth is decreased by 20%.

Height is increased by 30%.

Formula Used:

Volume of cuboid = Length × Breadth × Height

Calculation:

Let initial dimensions be L, B, and H.

Initial volume = L × B × H

New length = L × (1 - 0.10) = 0.90L

New breadth = B × (1 - 0.20) = 0.80B

New height = H × (1 + 0.30) = 1.30H

New volume = 0.90L × 0.80B × 1.30H

New volume = 0.90 × 0.80 × 1.30 × L × B × H

New volume = 0.936 × L × B × H

Percentage change in volume = -0.064 × 100

Percentage change in volume = -6.4%

Negative sign is indicating that volume will be decrese.

The total percentage change in the volume of the cuboid is 6.4%.

Given:

Diameter of the hemisphere = 14 cm

Radius of the hemisphere (r) = 14/2 = 7 cm

π = 22/7

Formula Used:

Volume of a hemisphere = (2/3) π r³

Calculation:

Volume of the hemisphere

Volume = (2/3) π r³

π = 22/7

r = 7 cm

→ Volume = (2/3) × (22/7) × 7³

→ Volume = (2/3) × (22/7) × 343

→ Volume = (2/3) × 22 × 49

→ Volume = (2 × 22 × 49)/3

→ Volume = 2156/3

The volume of the hemisphere is 2156/3 cm³.

Given:

Radius of the base and height of the cylinder are in the ratio 7 : 3.

Volume of the cylinder = 12474 cm³

Use π = 22/7

Formula Used:

Total surface area of the cylinder = 2πr(r + h)

Volume of the cylinder = πr²h

Calculation:

Let the radius of the base be 7x and the height be 3x.

Volume of the cylinder = πr²h

⇒ (22/7) × (7x)² × (3x) = 12474

⇒ (22/7) × 49x² × 3x = 12474

⇒ (22/7) × 147x³ = 12474

⇒ 22 × 21x³ = 12474

⇒ 462x³ = 12474

⇒ x³ = 27

⇒ x = 3

So, the radius of the base = 7x = 7 × 3 = 21 cm

Height of the cylinder = 3x = 3 × 3 = 9 cm

Total surface area of the cylinder = 2πr(r + h)

⇒ 2 × (22/7) × 21 × (21 + 9)

⇒ 2 × (22/7) × 21 × 30

⇒ 2 × 22 × 3 × 30

⇒ 2 × 22 × 90

⇒ 3960 cm²

The total surface area of the cylinder is 3960 cm².

Given:

Radius of the cone (r) = 7 units

Curved surface area (CSA) = 308 square units

π = 22/7

Formula used:

Curved surface area of a cone = πrℓ

Volume of a cone = (1/3)πr²h

Using Pythagoras theorem: ℓ² = r² + h²

Calculation:

Curved surface area = πrℓ

⇒ (22/7) × 7 × ℓ = 308

⇒ 22 × ℓ = 308

⇒ ℓ = 308 / 22 = 14 units

Using Pythagoras theorem to find height (h):

ℓ² = r² + h²

⇒ 14² = 7² + h²

⇒ 196 = 49 + h²

⇒ h² = 196 - 49

⇒ h² = 147

⇒ h = √147 = 7√3 units

Volume = (1/3)πr²h

⇒ Volume = (1/3) × (22/7) × 7² × 7√3

⇒ Volume = (1/3) × 22 × 49 × √3

⇒ Volume = (1/3) × 1078 × √3

⇒ Volume = 1079/√3

Given:

50% increase in the diagonal of the cube

Formula used:

Diagonal of a cube = √3 × side

Volume of a cube = side³

Calculation:

Let the sides of a cube be a cm each

volume = a³

a is increased by 50% becomes

a + (a/100) × 50 = a + a/2 = 3a/2

new volume =(3a/2)³ = 27a³/8

increase in volume =(27a3/8) - a³

=(27a³ – 8a³)/8 = 19a3/8

% of increase ={(19a³/8)×100}/a³

=19 × 25/2 = 475/2 = 237.5

Volume will be increased by 237.5%

Given:

Diameter of sphere is increased by 30%

Formula used:

Volume of sphere, V = (4/3) × π × r³

New radius, r_new = r × 1.30

New Volume, V_new = (4/3) × π × (r × 1.30)³

Calculation:

V_new = (4/3) × π × (r × 1.30)³

= (4/3) × π × r³ × 1.30³

= V × 2.197

Percentage increase in volume = (V_new - V) / V × 100%

= (2.197 - 1) × 100%

= 1.197 × 100%

= 119.7%

∴ The percentage increase in the volume is approximately 119.7%.

Length of edge of original cube = p cm

Volume of the original cube = p³

We know that a cube contains 8 corners therefore, 8 cubes are removed from the corners of the original cube.

Volume of removed cubes = 8 × (p/k)³

The original cube looses 12.5% of its volume.

Therefore, 8 × (p/k)³ = (1/8) × p³

⇒ k³ = 64

⇒ k = 4

Hence, the correct answer is 4.

Given:

Height of the cube = Height of the sphere = h

Formula Used:

Volume of the cube = a³, where a is the side length of the cube.

Volume of the sphere = (4/3) × π × r³, where r is the radius of the sphere.

Calculation:

Since the height of the sphere is equal to its diameter, we have:

h = 2r

So, r = h/2

For the cube, the height is equal to the side length:

a = h

Volume of the cube = a³ = h³

Volume of the sphere = (4/3) × π × (h/2)³

Volume of the sphere = (4/3) × π × (h³/8)

Volume of the sphere = (4πh³) / 24

Volume of the sphere = (πh³) / 6

Ratio of the volumes (cube : sphere) = h³ : (πh³ / 6)

⇒ Ratio = 6h³ / (πh³)

⇒ Ratio = 6 / π

The ratio of their volumes is 6 : π.

Given:

PQR is a right-angle triangle with ∠ Q = 90°,

PQ = 7 cm and QR = 24cm.

Formula used:

Pythagoras theorem 

(Hypotenuse)² = (Base)² +(Perpendicular)²

The volume of cone = (1/3)π r² h

Calculation:

Applying Pythagoras' theorem in triangle PQR

PR² = PQ² + QR²

⇒ PR2 = 72 + 242  

⇒ PP = √(49 + 576) =√ 625

⇒ PR = 25 cm 

Area of triangle PQR = (1/2) × Base × Height = (1/2) × PQ × QR = (1/2) × PR × OQ

⇒ OQ× 25 = 7× 24 

⇒ OQ = (7× 24)/25

The volume of double cone = volume of cone 1 + Volume of cone 2

The volume of the double cone = volume of PSQ + Volume of QRS

OQ = r =  (7× 24)/25

h₁ = PO 

h₂ = OR

h₁ + h₂ = PO + OR = PR = 25 cm

⇒ (1/3)π r² h₁ + (1/3)π r² h₂ = (1/3)π r² (h₁+h₂)

⇒ (1/3)(22/7) {(7× 24)/25}² × 25

⇒ (56× 24× 22)/25 = 29568/25

⇒ 1182.72 cm²

Hence, the correct option is (2).

Given:

Initial Length (L) = L

Initial Breadth (B) = B

Initial Height (H) = H

Length increased by 10%

Breadth increased by 20%

Height increased by 50%

Concept:

The volume of a cuboid is given by the product of its length, breadth, and height.

Formula Used:

Volume of cuboid = Length × Breadth × Height

Calculation:

Initial Volume (V_initial) = L × B × H

New Length (L_new) = L × (1 + 10/100) = 1.1L

New Breadth (B_new) = B × (1 + 20/100) = 1.2B

New Height (H_new) = H × (1 + 50/100) = 1.5H

New Volume (V_new) = L_new × B_new × H_new

V_new = 1.1L × 1.2B × 1.5H

V_new = 1.98LBH

Percentage Increase in Volume = [(V_new - V_initial)/V_initial] × 100

⇒ Percentage Increase = [(1.98LBH - LBH)/LBH] × 100

⇒ Percentage Increase = [0.98LBH/LBH] × 100

⇒ Percentage Increase = 0.98 × 100

⇒ Percentage Increase = 98%

⇒ Hence, the percentage increase in volume of the cuboid is 98%.