Two Figures MCQ Quiz - Objective Question with Answer for Two Figures - Download Free PDF

Calculation

ATQ, 2 ×(2πr) = 176

Or, 2πr = 88

Or, πr = 44

So, r = 14 cm

Volume of the cylinder = [22/7] r2h = 3528π

So, 196h = 3528

So, h = 18

Side of the square = 18 × [4 / 3] = 24 cm

Required area = 24 × 24 = 576 cm

Given:

Edge of solid iron cube = 24 cm

Thickness of rectangular sheet = 2 mm = 0.2 cm

Ratio of length (l) to breadth (b) = 6:5

Formula Used:

Volume of cube = Volume of rectangular sheet

Volume of cube = Edge³

Volume of rectangular sheet = Length × Breadth × Thickness

l / b = 6 / 5

Calculation:

Volume of cube = Edge³

⇒ Volume = 24³

⇒ Volume = 13824 cm³

Volume of rectangular sheet = l × b × 0.2

⇒ 13824 = l × b × 0.2

⇒ l × b = 13824 / 0.2

⇒ l × b = 69120

l / b = 6 / 5

⇒ l = 6k, b = 5k

⇒ l × b = 6k × 5k

⇒ 69120 = 30k²

⇒ k² = 69120 / 30

⇒ k² = 2304

⇒ k = √2304

⇒ k = 48

l = 6k = 6 × 48 = 288 cm

b = 5k = 5 × 48 = 240 cm

l + b = 288 + 240

⇒ l + b = 528 cm

The correct answer is option 2 (528 cm).

Given:

Radius of cylinder = 3 inches.

Height of cylinder = 8 inches.

Formula Used:

Volume of cylinder = πr²h

Volume of hemisphere = (2/3)πr³

Calculation:

Volume of cylinder = π × 3² × 8

⇒ Volume of cylinder = π × 9 × 8

⇒ Volume of cylinder = 72π cubic inches

Volume of hemisphere = (2/3)π × 3³

⇒ Volume of hemisphere = (2/3)π × 27

⇒ Volume of hemisphere = 18π cubic inches

Number of hemispheres = Volume of cylinder / Volume of hemisphere

⇒ Number of hemispheres = 72π / 18π

⇒ Number of hemispheres = 4

The number of hemispheres formed is 4.

Given:

Edge of the wooden cube = 14 cm.

Formula Used:

Volume of the cube = Edge³

Volume of the cone = (1/3) × π × r² × h

Maximum volume of the cone when it is carved from a cube: r = h/2, where h = edge of the cube.

Volume of the removed portion = Volume of the cube - Volume of the cone.

Calculation:

Volume of the cube = Edge³

⇒ Volume of the cube = 14³

⇒ Volume of the cube = 2744 cm³

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

Height (h) of the cone = 14 cm

Volume of the cone = (1/3) × π × r² × h

⇒ Volume of the cone = (1/3) × 22/7 × 7² × 14

⇒ Volume of the cone = (1/3) × 22 × 49 × 2

⇒ Volume of the cone = (1/3) × 2156

⇒ Volume of the cone = 718.67 cm³

Volume of the removed portion (V) = Volume of the cube - Volume of the cone

⇒ V = 2744 - 718.67

⇒ V = 2025.33 cm³

3V = 3 × 2025.33

⇒ 3V = 6076 cm³

The value of 3V is 6076 cm³.

Given:

Height of cone = 13 cm

Radius of cone = 4 cm

Radius of hemisphere = 4 cm

Formula Used:

Volume of cone = (1/3) × π × r² × h

Volume of hemisphere = (2/3) × π × r³

Total volume = Volume of cone + Volume of hemisphere

Calculation:

Volume of cone = (1/3) × π × 4² × 13

⇒ Volume of cone = (1/3) × π × 16 × 13

⇒ Volume of cone = (208/3)π

Volume of hemisphere = (2/3) × π × 4³

⇒ Volume of hemisphere = (2/3) × π × 64

⇒ Volume of hemisphere = (128/3)π

Total volume = (208/3)π + (128/3)π

⇒ Total volume = (336/3)π

⇒ Total volume = 112π

Using π ≈ 3.14:

Total volume = 112 × 3.14

⇒ Total volume = 351.68 cm³

The volume of the ice-cream is approximately 352 cm³.

Given:

Radius of Sphere = 42 cm

Radius of wire = 21 cm

Formula:

Volume of cylinder = πr²h

Volume of sphere = [4/3]πr³

Calculation:

Let length of the wire be x, then

According to the question

π × 21 × 21 × x = [4/3] × π × 42 × 42 × 42 [As volume will remain constant]

⇒ x = (4 × 42 × 42 × 42)/(21 × 21 × 3)

⇒ x = 224 cm 

∴ The length of the wire is 224 cm

Given:

Radius of sphere = 12 cm

Height of cone = 12 cm

Formula:

Volume of cone = (1/3) × πr²h

Volume of sphere = (4/3) × πr³

Calculation:

Let radius of the cone be r cm

According to the question

(1/3) × π × r² × 12 = (4/3) × π × 12 × 12 × 12

⇒ r² = 12 × 12 × 4

⇒ r = 12 × 2

FORMULA USED:

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

Here l, b and h are length, breadth and height.

Volume of cube = a³

CALCULATION:

a³ = 729

⇒ a = 9 cm.

Length of cuboid = 9 + 9 = 18 cm.

Breadth = 9 cm

Height = 9 cm

Total surface area of cuboid = 2 (18 × 9 + 9 × 9 + 9 × 18) = 810 cm²

∴ Total surface area of the cuboid is 810 cm²

GIVEN:

Radius of sphere = 3 cm

Height of the cone = Half of the radius of the cone

CONCEPT:

Volume of Sphere = Volume of Cone

FORMULA USED:

Volume of Sphere = 4/3 × πR³

Volume of Cone = 1/3 × πr²h

CALCULATION:

Suppose the height and radius of the cone are ‘h’ and ‘r’ respectively.

∴ h = r/2

Now,

Applying the formula:

\(\frac{4}{3} \times \pi \times 3 \times 3 \times 3 = \frac{1}{3} \times \pi \times r \times r \times h\)

Put h = r/2

⇒ \(\frac{4}{3} \times 3 \times 3 \times 3 = \frac{1}{3} \times r \times r \times \frac{r}{2}\)

⇒ r³ = 216

⇒ r = 6

∴ Radius of the cone = 6 cm.

Given:

A glass cylinder with diameter 20 cm has water to a height of 9 cm. A metal cube of 8 cm edge is immersed in it completely.

Formula used:

Volume of cylinder = Πr²h

Volume of cube = a³ 

Calculation:

Diameter of cylinder = 20 cm

⇒ Radius of cylinder = 10 cm

Now, Volume of water displaced (because of which water will rise in cylinder) = Volume of cube

∴ πr²h = a3

⇒ 3.142 × 10 × 10 × h = 8³

⇒ 3142 × 1/10 × h = 512

⇒ h = 5120/3142 

⇒ h = 1.62 cm ~ 1.6 cm

Given:

Height of cylinder = 30 cm

Radius of cylinder = 5 cm

Height of cone = 12 cm

Radius of cone = 5 cm

Formula used:

The surface area of cylinder = 2πrh

The surface area of cone = πrl

l² = h² + r²

Where,

l = slant height of the cone

h = height

r = radius

Calculation:

l² = h² + r²

⇒ l² = 12² + 5²

⇒ l² = 144 + 25

⇒ l = 13 cm

The surface area of the remaining figure = surface area of cylinder + 2 × Curved surface area of the cone

⇒ 2πrh + 2πrl

⇒ 2πr(h + l)

⇒ 2π × 5(30 + 13)

⇒ 430π

∴ The surface area of the remaining solid is 430π.

 Additional Information

When the cones are drilled out then the volume of a cylinder is decreasing. But the surface area will increase. Surface area means the area which we can touch.  When the cones drilled out then we can touch both outer and the inner surface. So we have to add both surface areas. 

Given:

Cube of side 20 cm is recast into a cuboid of length and breadth 40cm respectively. 

Formulas used:

Volume of cube = (side)³ 

Volume of cuboid = l × b × h 

Diagonal of cuboid = √ l² + b² + h²

Calculation:

Volume of cube = Volume of cuboid 

⇒ 20 × 20 × 20 = 40 × 40 × h

⇒ h = (20 × 20 × 20) ÷ (40 × 40)

⇒ h = 5 cm 

Diagonal of the cuboid = √ 40² + 40² + 5²

⇒ √ 1600 + 1600 + 25 = √ 3225

⇒ 5√129 cm

∴ The diagonal of the cuboid = 5√129 cm

Given:

The radius of the sphere is 3 cm

Radius & height of cylinder are 4.5 cm & 32cm

Formula used:

The volume of sphere = 4/3πr³

The volume of cylinder = πr²h

Calculations:

The volume of sphere inserted = volume of water risen

Radius of sphere = 6/2= 3 cm

Radius of cylinder = 9/2 = 4.5

So,

(4/3)π × (3)³ × Number of sphere =  π (4.5) × (4.5) × 32

⇒ Number of sphere = 18

∴ The correct choice is option 2.

Given:

Side of cube = 21 cm 

Formulas used:

Volume of sphere = 4/3πr³

Calculation:

The largest sphere that can be carved out of a cube of side 21 cm will have the diameter equal to 21 cm. 

Radius of sphere = 21/2 cm 

Volume of sphere = 4/3 × 22/7 × 21/2 × 21/2 × 21/2 

⇒ 11 × 21 × 21 

⇒ 4851 cm³

∴ The required result is 4851 cm3.

Given :

Radius of right circular cone and cylinder(r) = 5 cm

Height of right circular cone and cylinder(h) = 9 cm

Formula used :

Volume of right circular cylinder(V₁) = πr²h

The volume of the right circular cone (V₂) = \(\dfrac{1}{3}\)πr2h

Calculation :

Volume of remaining solid = V₁ - V₂ 

⇒ \(π× 5^2× 9 - \dfrac{1}{3}× π× 5^2× 9\) 

⇒ π × 5² × 9 (1 - \(\dfrac{1}{3}\))

⇒ \(\dfrac{2}{3}× π× 5^2× 9\)

⇒ 150π

∴ The answer is 150π .