Mensuration- 1 Mark Solutions
Important Mensuration Formulas:
• Volume of Cylinder = πr²h
• Volume of Cone = (1/3)πr²h
• Volume of Sphere = (4/3)πr³
• Volume of Hemisphere = (2/3)πr³
• Curved Surface Area of Cylinder = 2πrh
• Curved Surface Area of Cone = πrl (l = slant height)
• Surface Area of Hemisphere = 3πr²
• Relationship: Volume of Cone = (1/3) × Volume of Cylinder (same base and height)
• Volume of Cone = (1/3)πr²h
• Volume of Sphere = (4/3)πr³
• Volume of Hemisphere = (2/3)πr³
• Curved Surface Area of Cylinder = 2πrh
• Curved Surface Area of Cone = πrl (l = slant height)
• Surface Area of Hemisphere = 3πr²
• Relationship: Volume of Cone = (1/3) × Volume of Cylinder (same base and height)
1
If a cylinder and a cone are of the same radius and height, then how many cones full of milk can fill the cylinder? Answer with reasons.
(M'15)
Step 1: Write the volume formulas
Volume of Cylinder = πr²h
Volume of Cone = (1/3)πr²h
Volume of Cone = (1/3)πr²h
Step 2: Compare the volumes
Volume of Cylinder / Volume of Cone = (πr²h) / ((1/3)πr²h) = 3
∴ 3 cones full of milk can fill the cylinder.
2
If the radius of hemisphere is 21cm, then find its volume.
(J'15)
Step 1: Write the volume formula
Volume of Hemisphere = (2/3)πr³
Step 2: Substitute values
r = 21 cm
Volume = (2/3) × (22/7) × (21)³
Volume = (2/3) × (22/7) × (21)³
Step 3: Calculate
Volume = (2/3) × (22/7) × 9261
= (2/3) × 22 × 1323
= (2/3) × 29106
= 19404 cm³
= (2/3) × 22 × 1323
= (2/3) × 29106
= 19404 cm³
Calculate Hemisphere Volume
∴ The volume of the hemisphere is 19404 cm³.
3
"A conical solid block is exactly fitted inside the cubical box of side 'a', then the volume of conical solid block is 4/3 π a³". Is this statement true? Justify your answer.
(M'16)
Step 1: Analyze the dimensions
If a cone is exactly fitted inside a cube of side 'a':
- Height of cone (h) = a
- Radius of cone (r) = a/2
- Height of cone (h) = a
- Radius of cone (r) = a/2
Step 2: Calculate actual volume
Volume of Cone = (1/3)πr²h = (1/3)π(a/2)²(a) = (1/3)π(a²/4)(a) = (1/12)πa³
Step 3: Compare with given statement
Given statement says volume = (4/3)πa³
Actual volume = (1/12)πa³
(4/3)πa³ ≠ (1/12)πa³
Actual volume = (1/12)πa³
(4/3)πa³ ≠ (1/12)πa³
∴ The statement is false. The correct volume is (1/12)πa³, not (4/3)πa³.
4
If the surface area of a hemisphere is 'S', then express 'r' in terms of 'S'.
(M'16)
Step 1: Write the surface area formula
Surface Area of Hemisphere = 3πr²
Step 2: Set up equation
S = 3πr²
Step 3: Solve for r
r² = S/(3π)
r = √(S/(3π))
r = √(S/(3π))
∴ r = √(S/(3π))
5
Find the curved surface area of a cylinder of radius 14cm and height 21cm. (π = 22/7)
(J'16)
Step 1: Write the formula
Curved Surface Area of Cylinder = 2πrh
Step 2: Substitute values
r = 14 cm, h = 21 cm, π = 22/7
CSA = 2 × (22/7) × 14 × 21
CSA = 2 × (22/7) × 14 × 21
Step 3: Calculate
CSA = 2 × 22 × 2 × 21 = 2 × 22 × 42 = 1848 cm²
Calculate Cylinder CSA
∴ The curved surface area is 1848 cm².
6
Write the formula to find curved surface area of a cone and explain each term in it.
(M'17)
Formula:
Curved Surface Area of Cone = πrl
Explanation of terms:
π (pi) = Mathematical constant, approximately 3.14159
r = Radius of the base of the cone
l = Slant height of the cone = √(r² + h²) where h is the height
r = Radius of the base of the cone
l = Slant height of the cone = √(r² + h²) where h is the height
∴ The curved surface area of a cone is πrl, where r is the radius and l is the slant height.
7
If a cone is inscribed in a cylinder, what is the ratio of their volumes?
(J'17)
Step 1: Understand the relationship
When a cone is inscribed in a cylinder:
- They have the same base radius (r)
- They have the same height (h)
- They have the same base radius (r)
- They have the same height (h)
Step 2: Write volume formulas
Volume of Cylinder = πr²h
Volume of Cone = (1/3)πr²h
Volume of Cone = (1/3)πr²h
Step 3: Find the ratio
Volume of Cone : Volume of Cylinder = (1/3)πr²h : πr²h = 1:3
∴ The ratio of their volumes is 1:3.
8
The vertex angle of a cone is 60°. Find the ratio of the diameter with the height of the cone.
(J'17)
Step 1: Understand the geometry
Vertex angle = 60° means the angle at the apex of the cone is 60°.
This creates an equilateral triangle in the vertical cross-section.
This creates an equilateral triangle in the vertical cross-section.
Step 2: Relate diameter and height
In the cross-section, we have an equilateral triangle:
- Slant height (l) = diameter (d)
- Height (h) = (√3/2) × l = (√3/2) × d
- Slant height (l) = diameter (d)
- Height (h) = (√3/2) × l = (√3/2) × d
Step 3: Find the ratio
d : h = d : (√3/2)d = 1 : (√3/2) = 2 : √3
∴ The ratio of diameter to height is 2:√3.
9
"Cuboid is one of right prism". Is it true? Justify.
(J'17)
Step 1: Define right prism
A right prism is a prism in which the lateral faces are perpendicular to the bases.
Step 2: Analyze cuboid
A cuboid has:
- Rectangular bases
- Lateral faces perpendicular to the bases
- Rectangular bases
- Lateral faces perpendicular to the bases
∴ Yes, the statement is true. A cuboid is a special case of a right prism where all faces are rectangles.
10
Write the formula to find the volume of a cone and explain each term in it.
(J'18)
Formula:
Volume of Cone = (1/3)πr²h
Explanation of terms:
π (pi) = Mathematical constant, approximately 3.14159
r = Radius of the base of the cone
h = Height of the cone (perpendicular distance from apex to base)
r = Radius of the base of the cone
h = Height of the cone (perpendicular distance from apex to base)
∴ The volume of a cone is (1/3)πr²h, where r is the radius and h is the height.
11
Find the value of liquid hemispherical bowl can hold, where radius of the ball is 4.2 cm.
(J'18)
Step 1: Write the volume formula
Volume of Hemisphere = (2/3)πr³
Step 2: Substitute values
r = 4.2 cm, π = 22/7
Volume = (2/3) × (22/7) × (4.2)³
Volume = (2/3) × (22/7) × (4.2)³
Step 3: Calculate
(4.2)³ = 4.2 × 4.2 × 4.2 = 74.088
Volume = (2/3) × (22/7) × 74.088
= (2/3) × 22 × 10.584
= (2/3) × 232.848
= 155.232 cm³
Volume = (2/3) × (22/7) × 74.088
= (2/3) × 22 × 10.584
= (2/3) × 232.848
= 155.232 cm³
Calculate Hemisphere Volume
∴ The hemispherical bowl can hold 155.232 cm³ of liquid.
12
In a hemispherical bowl of 2.1 cm radius ice-cream is there. Find the volume of the bowl.
(M'19)
Step 1: Write the volume formula
Volume of Hemisphere = (2/3)πr³
Step 2: Substitute values
r = 2.1 cm, π = 22/7
Volume = (2/3) × (22/7) × (2.1)³
Volume = (2/3) × (22/7) × (2.1)³
Step 3: Calculate
(2.1)³ = 2.1 × 2.1 × 2.1 = 9.261
Volume = (2/3) × (22/7) × 9.261
= (2/3) × 22 × 1.323
= (2/3) × 29.106
= 19.404 cm³
Volume = (2/3) × (22/7) × 9.261
= (2/3) × 22 × 1.323
= (2/3) × 29.106
= 19.404 cm³
∴ The volume of the hemispherical bowl is 19.404 cm³.
13
If the metallic cylinder of height 4 cm and radius 3 cm is melted under recast into a sphere, then find the radius of the sphere.
(J'19)
Step 1: Write volume formulas
Volume of Cylinder = πr²h
Volume of Sphere = (4/3)πR³
Volume of Sphere = (4/3)πR³
Step 2: Set volumes equal
Since the material is recast, volumes are equal:
πr²h = (4/3)πR³
πr²h = (4/3)πR³
Step 3: Substitute values and solve
π × 3² × 4 = (4/3)πR³
π × 9 × 4 = (4/3)πR³
36π = (4/3)πR³
36 = (4/3)R³
R³ = 36 × (3/4) = 27
R = ∛27 = 3 cm
π × 9 × 4 = (4/3)πR³
36π = (4/3)πR³
36 = (4/3)R³
R³ = 36 × (3/4) = 27
R = ∛27 = 3 cm
Calculate Sphere Radius
∴ The radius of the sphere is 3 cm.
14
Write the formula for finding lateral surface area of a cylinder and explain each term in it.
(J'19)
Formula:
Lateral Surface Area of Cylinder = 2πrh
Explanation of terms:
π (pi) = Mathematical constant, approximately 3.14159
r = Radius of the base of the cylinder
h = Height of the cylinder
r = Radius of the base of the cylinder
h = Height of the cylinder
∴ The lateral surface area of a cylinder is 2πrh, where r is the radius and h is the height.
15
A joker cap is in the form of a right circular cone, whose base radius is 7 cm and slant height is 25 cm. Find it's curved surface area.
(May 2022)
Step 1: Write the formula
Curved Surface Area of Cone = πrl
Step 2: Substitute values
r = 7 cm, l = 25 cm, π = 22/7
CSA = (22/7) × 7 × 25
CSA = (22/7) × 7 × 25
Step 3: Calculate
CSA = 22 × 25 = 550 cm²
Calculate Cone CSA
∴ The curved surface area of the joker cap is 550 cm².
16
If the ratio of a base radii of two right circular cylinder is 1:2 and the ratio of their heights is 2:3 then find the ratio of their volumes.
(Aug.22)
Step 1: Write the volume formula
Volume of Cylinder = πr²h
Step 2: Set up ratios
Let r₁, h₁ be radius and height of first cylinder
Let r₂, h₂ be radius and height of second cylinder
r₁:r₂ = 1:2 ⇒ r₁ = k, r₂ = 2k
h₁:h₂ = 2:3 ⇒ h₁ = 2m, h₂ = 3m
Let r₂, h₂ be radius and height of second cylinder
r₁:r₂ = 1:2 ⇒ r₁ = k, r₂ = 2k
h₁:h₂ = 2:3 ⇒ h₁ = 2m, h₂ = 3m
Step 3: Calculate volume ratio
V₁ = πr₁²h₁ = π(k)²(2m) = 2πk²m
V₂ = πr₂²h₂ = π(2k)²(3m) = π(4k²)(3m) = 12πk²m
V₁:V₂ = 2πk²m : 12πk²m = 2:12 = 1:6
V₂ = πr₂²h₂ = π(2k)²(3m) = π(4k²)(3m) = 12πk²m
V₁:V₂ = 2πk²m : 12πk²m = 2:12 = 1:6
∴ The ratio of their volumes is 1:6.
Mensuration Problems (2 Marks) - Complete Solutions
Important Mensuration Formulas:
• Volume of Sphere = (4/3)πr³
• Surface Area of Sphere = 4πr²
• Volume of Cube = a³
• Curved Surface Area of Cone = πrl (l = slant height)
• Volume of Cone = (1/3)πr²h
• Curved Surface Area of Cylinder = 2πrh
• Volume of Cylinder = πr²h
• Volume of Hemisphere = (2/3)πr³
• Surface Area of Sphere = 4πr²
• Volume of Cube = a³
• Curved Surface Area of Cone = πrl (l = slant height)
• Volume of Cone = (1/3)πr²h
• Curved Surface Area of Cylinder = 2πrh
• Volume of Cylinder = πr²h
• Volume of Hemisphere = (2/3)πr³
1
The radius of a spherical balloon increases from 7cm to 14 cm as air pumped into it. Find the ratio of the volumes of the balloon before and after pumping the air.
(M'15)
Step 1: Write the volume formula for sphere
Volume of Sphere = (4/3)πr³
Step 2: Calculate volumes
Initial Volume (r = 7 cm) = (4/3)π(7)³
Final Volume (r = 14 cm) = (4/3)π(14)³
Final Volume (r = 14 cm) = (4/3)π(14)³
Step 3: Find the ratio
Ratio = Initial Volume : Final Volume
= [(4/3)π(7)³] : [(4/3)π(14)³]
= (7)³ : (14)³
= 343 : 2744
= 1 : 8 (dividing by 343)
= [(4/3)π(7)³] : [(4/3)π(14)³]
= (7)³ : (14)³
= 343 : 2744
= 1 : 8 (dividing by 343)
Calculate Volume Ratio
∴ The ratio of the volumes is 1:8.
2
Find the volume and surface area of a sphere of radius 42cm (π = 22/7)
(M'16)
Step 1: Write the formulas
Volume of Sphere = (4/3)πr³
Surface Area of Sphere = 4πr²
Surface Area of Sphere = 4πr²
Step 2: Calculate volume
r = 42 cm, π = 22/7
Volume = (4/3) × (22/7) × (42)³
= (4/3) × (22/7) × 74088
= (4/3) × 22 × 10584
= (4/3) × 232848
= 310464 cm³
Volume = (4/3) × (22/7) × (42)³
= (4/3) × (22/7) × 74088
= (4/3) × 22 × 10584
= (4/3) × 232848
= 310464 cm³
Step 3: Calculate surface area
Surface Area = 4 × (22/7) × (42)²
= 4 × (22/7) × 1764
= 4 × 22 × 252
= 4 × 5544
= 22176 cm²
= 4 × (22/7) × 1764
= 4 × 22 × 252
= 4 × 5544
= 22176 cm²
Calculate Sphere Properties
∴ Volume = 310464 cm³, Surface Area = 22176 cm².
3
A solid metallic ball of volume 64cm³ melted and made into a solid cube. Find the side of the solid cube.
(M'16)
Step 1: Understand the concept
When a solid is melted and recast, its volume remains the same.
Step 2: Write the volume formulas
Volume of Sphere = 64 cm³
Volume of Cube = a³ (where a is the side)
Volume of Cube = a³ (where a is the side)
Step 3: Equate volumes and solve
a³ = 64
a = ∛64 = 4 cm
a = ∛64 = 4 cm
Calculate Cube Side
∴ The side of the solid cube is 4 cm.
4
A toy is in the form of a cone mounted on a hemisphere. The radius of the base and the height of the cone are 7cm and 8cm respectively. Find the surface area of the toy.
(J'16)
Step 1: Understand the shape
The toy consists of:
- A cone with radius 7 cm and height 8 cm
- A hemisphere with radius 7 cm
Surface area of toy = CSA of cone + CSA of hemisphere
- A cone with radius 7 cm and height 8 cm
- A hemisphere with radius 7 cm
Surface area of toy = CSA of cone + CSA of hemisphere
Step 2: Calculate slant height of cone
l = √(r² + h²) = √(7² + 8²) = √(49 + 64) = √113 ≈ 10.63 cm
Step 3: Calculate surface areas
CSA of Cone = πrl = (22/7) × 7 × √113 = 22 × √113 ≈ 22 × 10.63 = 233.86 cm²
CSA of Hemisphere = 2πr² = 2 × (22/7) × 7² = 2 × 22 × 7 = 308 cm²
Total Surface Area = 233.86 + 308 = 541.86 cm²
CSA of Hemisphere = 2πr² = 2 × (22/7) × 7² = 2 × 22 × 7 = 308 cm²
Total Surface Area = 233.86 + 308 = 541.86 cm²
Calculate Toy Surface Area
∴ The surface area of the toy is approximately 541.86 cm².
5
The diameter of a solid sphere is 6 cm. It is melted and recast into a solid cylinder of height 4 cm. Find the radius of cylinder.
(M'17)
Step 1: Understand the concept
When a solid is melted and recast, its volume remains the same.
Step 2: Write the volume formulas
Diameter of sphere = 6 cm, so radius = 3 cm
Volume of Sphere = (4/3)πr³ = (4/3)π(3)³ = (4/3)π × 27 = 36π cm³
Volume of Cylinder = πR²h (where R is radius of cylinder, h = 4 cm)
Volume of Sphere = (4/3)πr³ = (4/3)π(3)³ = (4/3)π × 27 = 36π cm³
Volume of Cylinder = πR²h (where R is radius of cylinder, h = 4 cm)
Step 3: Equate volumes and solve
πR² × 4 = 36π
R² × 4 = 36
R² = 9
R = 3 cm
R² × 4 = 36
R² = 9
R = 3 cm
Calculate Cylinder Radius
∴ The radius of the cylinder is 3 cm.
6
The height and the base radius of a Cone and a Cylinder are equal to the radius of a Sphere. Find the ratio of their volumes.
(M'18)
Step 1: Define the variables
Let the common radius = r
Then:
- Height of cone = r
- Height of cylinder = r
- Radius of sphere = r
Then:
- Height of cone = r
- Height of cylinder = r
- Radius of sphere = r
Step 2: Write the volume formulas
Volume of Cone = (1/3)πr²h = (1/3)πr²(r) = (1/3)πr³
Volume of Cylinder = πr²h = πr²(r) = πr³
Volume of Sphere = (4/3)πr³
Volume of Cylinder = πr²h = πr²(r) = πr³
Volume of Sphere = (4/3)πr³
Step 3: Find the ratio
Cone : Cylinder : Sphere = (1/3)πr³ : πr³ : (4/3)πr³
= (1/3) : 1 : (4/3)
Multiply by 3: 1 : 3 : 4
= (1/3) : 1 : (4/3)
Multiply by 3: 1 : 3 : 4
∴ The ratio of volumes (Cone : Cylinder : Sphere) is 1 : 3 : 4.
7
The diameter of the base of a right circular cone is 12 cm and volume 376.8 cm³. Find its height (π = 3.14)
(J'18)
Step 1: Write the volume formula
Volume of Cone = (1/3)πr²h
Step 2: Substitute known values
Diameter = 12 cm, so radius r = 6 cm
Volume = 376.8 cm³, π = 3.14
376.8 = (1/3) × 3.14 × (6)² × h
Volume = 376.8 cm³, π = 3.14
376.8 = (1/3) × 3.14 × (6)² × h
Step 3: Solve for height
376.8 = (1/3) × 3.14 × 36 × h
376.8 = (1/3) × 113.04 × h
376.8 = 37.68 × h
h = 376.8 / 37.68 = 10 cm
376.8 = (1/3) × 113.04 × h
376.8 = 37.68 × h
h = 376.8 / 37.68 = 10 cm
Calculate Cone Height
∴ The height of the cone is 10 cm.
8
A right circular cylinder has radius 3.5 cm and height 14 cm. Find curved surface area.
(M'19)
Step 1: Write the formula
Curved Surface Area of Cylinder = 2πrh
Step 2: Substitute values
r = 3.5 cm, h = 14 cm, π = 22/7
CSA = 2 × (22/7) × 3.5 × 14
CSA = 2 × (22/7) × 3.5 × 14
Step 3: Calculate
CSA = 2 × 22 × 0.5 × 14
= 2 × 22 × 7
= 44 × 7 = 308 cm²
= 2 × 22 × 7
= 44 × 7 = 308 cm²
Calculate Cylinder CSA
∴ The curved surface area of the cylinder is 308 cm².
Mensuration Problems (4 Marks) - Complete Solutions
Important Mensuration Formulas:
• Volume of Cylinder = πr²h
• Volume of Sphere = (4/3)πr³
• Volume of Hemisphere = (2/3)πr³
• Volume of Cone = (1/3)πr²h
• Volume of Cuboid = l × b × h
• Volume of Cube = a³
• Curved Surface Area of Cone = πrl (l = slant height)
• Total Surface Area of Cylinder = 2πr(h + r)
• Lateral Surface Area of Cube = 4a²
• Area of Sector = (θ/360) × πr²
• Volume of Sphere = (4/3)πr³
• Volume of Hemisphere = (2/3)πr³
• Volume of Cone = (1/3)πr²h
• Volume of Cuboid = l × b × h
• Volume of Cube = a³
• Curved Surface Area of Cone = πrl (l = slant height)
• Total Surface Area of Cylinder = 2πr(h + r)
• Lateral Surface Area of Cube = 4a²
• Area of Sector = (θ/360) × πr²
1
A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. If the length of the cylindrical part of the capsule is 14mm and the diameter of hemisphere is 6mm, then find the volume of medicine capsule.
(M'15)
Step 1: Understand the shape
The capsule consists of:
- A cylinder of height 14 mm and radius 3 mm
- Two hemispheres of radius 3 mm each
- A cylinder of height 14 mm and radius 3 mm
- Two hemispheres of radius 3 mm each
Step 2: Calculate volume of cylinder
Volume of Cylinder = πr²h = π × (3)² × 14 = π × 9 × 14 = 126π mm³
Step 3: Calculate volume of two hemispheres
Volume of one hemisphere = (2/3)πr³ = (2/3)π × (3)³ = (2/3)π × 27 = 18π mm³
Volume of two hemispheres = 2 × 18π = 36π mm³
Volume of two hemispheres = 2 × 18π = 36π mm³
Step 4: Calculate total volume
Total Volume = Volume of Cylinder + Volume of two hemispheres
= 126π + 36π = 162π mm³
Using π = 22/7: 162 × (22/7) = 3564/7 = 509.14 mm³
= 126π + 36π = 162π mm³
Using π = 22/7: 162 × (22/7) = 3564/7 = 509.14 mm³
Calculate Capsule Volume
∴ The volume of the medicine capsule is 162π mm³ ≈ 509.14 mm³.
2
The area of a sector-shaped canvas cloth is 264m². With this canvas cloth, if a right circular conical tent is erected with the radius of the base as 7m, then find the height of the tent.
(J'15)
Step 1: Understand the relationship
The area of the sector equals the curved surface area of the cone.
Curved Surface Area of Cone = πrl
Curved Surface Area of Cone = πrl
Step 2: Find slant height
Given: CSA = 264 m², r = 7 m, π = 22/7
264 = (22/7) × 7 × l
264 = 22 × l
l = 264 / 22 = 12 m
264 = (22/7) × 7 × l
264 = 22 × l
l = 264 / 22 = 12 m
Step 3: Find height of cone
Using Pythagoras theorem: h = √(l² - r²) = √(12² - 7²) = √(144 - 49) = √95 ≈ 9.75 m
Calculate Cone Height
∴ The height of the tent is √95 m ≈ 9.75 m.
3
DWACRA is supplied cuboidal shaped wax block with measurements 88cm x 42cm x 35cm. From this how many number of cylindrical candles of 2.8cm diameter and 8cm of height can be prepared?
(M'16)
Step 1: Calculate volume of wax block
Volume of Cuboid = l × b × h = 88 × 42 × 35 = 129360 cm³
Step 2: Calculate volume of one candle
Diameter = 2.8 cm, so radius = 1.4 cm, height = 8 cm
Volume of Cylinder = πr²h = (22/7) × (1.4)² × 8
= (22/7) × 1.96 × 8 = (22/7) × 15.68 = 49.28 cm³
Volume of Cylinder = πr²h = (22/7) × (1.4)² × 8
= (22/7) × 1.96 × 8 = (22/7) × 15.68 = 49.28 cm³
Step 3: Calculate number of candles
Number of candles = Volume of wax block / Volume of one candle
= 129360 / 49.28 ≈ 2625
= 129360 / 49.28 ≈ 2625
Calculate Number of Candles
∴ 2625 cylindrical candles can be prepared from the wax block.
4
How many spherical balls each 7cm in diameter can be made out of a solid lead cube whose edge measures 66cm?
(J'16)
Step 1: Calculate volume of cube
Volume of Cube = a³ = 66³ = 287496 cm³
Step 2: Calculate volume of one ball
Diameter = 7 cm, so radius = 3.5 cm
Volume of Sphere = (4/3)πr³ = (4/3) × (22/7) × (3.5)³
= (4/3) × (22/7) × 42.875 = (4/3) × 22 × 6.125 = (4/3) × 134.75 ≈ 179.67 cm³
Volume of Sphere = (4/3)πr³ = (4/3) × (22/7) × (3.5)³
= (4/3) × (22/7) × 42.875 = (4/3) × 22 × 6.125 = (4/3) × 134.75 ≈ 179.67 cm³
Step 3: Calculate number of balls
Number of balls = Volume of cube / Volume of one ball
= 287496 / 179.67 ≈ 1600
= 287496 / 179.67 ≈ 1600
Calculate Number of Balls
∴ 1600 spherical balls can be made from the lead cube.
5
The length of a cuboid is 12 cm, breadth and height are equal in measurements, and its volume is 432 cm³. The cuboid is cut into two cubes. Find the lateral surface area of each cube.
(M'17)
Step 1: Find breadth and height
Let breadth = height = x cm
Volume = l × b × h = 12 × x × x = 432
12x² = 432
x² = 36
x = 6 cm
Volume = l × b × h = 12 × x × x = 432
12x² = 432
x² = 36
x = 6 cm
Step 2: Understand the cutting
Cuboid dimensions: 12 cm × 6 cm × 6 cm
When cut into two cubes, each cube will have side = 6 cm
When cut into two cubes, each cube will have side = 6 cm
Step 3: Calculate lateral surface area of one cube
Lateral Surface Area of Cube = 4a² = 4 × (6)² = 4 × 36 = 144 cm²
Calculate Cube Properties
∴ The lateral surface area of each cube is 144 cm².
6
How many silver coins of diameter 5 cm and thickness 4 mm have to be melted to prepare a cuboid of 12 cm × 11 cm × 5 cm dimension?
(M'18)
Step 1: Calculate volume of cuboid
Volume of Cuboid = l × b × h = 12 × 11 × 5 = 660 cm³
Step 2: Calculate volume of one coin
Diameter = 5 cm, so radius = 2.5 cm
Thickness = 4 mm = 0.4 cm
Volume of Coin (cylinder) = πr²h = (22/7) × (2.5)² × 0.4
= (22/7) × 6.25 × 0.4 = (22/7) × 2.5 ≈ 7.857 cm³
Thickness = 4 mm = 0.4 cm
Volume of Coin (cylinder) = πr²h = (22/7) × (2.5)² × 0.4
= (22/7) × 6.25 × 0.4 = (22/7) × 2.5 ≈ 7.857 cm³
Step 3: Calculate number of coins
Number of coins = Volume of cuboid / Volume of one coin
= 660 / 7.857 ≈ 84
= 660 / 7.857 ≈ 84
Calculate Number of Coins
∴ 84 silver coins are needed to make the cuboid.
7
A metallic sphere of diameter 30 cm is melted and recast into a cylinder of radius 10 cm. Find the height of the cylinder.
(J'18)
Step 1: Calculate volume of sphere
Diameter = 30 cm, so radius = 15 cm
Volume of Sphere = (4/3)πr³ = (4/3)π(15)³ = (4/3)π × 3375 = 4500π cm³
Volume of Sphere = (4/3)πr³ = (4/3)π(15)³ = (4/3)π × 3375 = 4500π cm³
Step 2: Calculate height of cylinder
Volume of Cylinder = πr²h = π × (10)² × h = 100πh cm³
Since volumes are equal: 100πh = 4500π
h = 4500π / 100π = 45 cm
Since volumes are equal: 100πh = 4500π
h = 4500π / 100π = 45 cm
Calculate Cylinder Height
∴ The height of the cylinder is 45 cm.
8
A toy is made with seven equal cubes of sides √7 cm. Six cubes are joined to six faces of a seventh cube. Find the total surface area of the toy.
(M'19)
Step 1: Understand the arrangement
One central cube with six cubes attached to each of its faces.
Side of each cube = √7 cm
Surface area of one cube = 6a² = 6 × (√7)² = 6 × 7 = 42 cm²
Side of each cube = √7 cm
Surface area of one cube = 6a² = 6 × (√7)² = 6 × 7 = 42 cm²
Step 2: Calculate visible surfaces
Central cube: 6 faces, but each face is covered by another cube, so no surface visible
Each attached cube: 5 faces visible (one face attached to central cube)
Total visible faces = 6 cubes × 5 faces = 30 faces
Each attached cube: 5 faces visible (one face attached to central cube)
Total visible faces = 6 cubes × 5 faces = 30 faces
Step 3: Calculate total surface area
Area of one face = a² = (√7)² = 7 cm²
Total Surface Area = 30 × 7 = 210 cm²
Total Surface Area = 30 × 7 = 210 cm²
Calculate Toy Surface Area
∴ The total surface area of the toy is 210 cm².
9
A cylindrical tank of radius 7 m has water to some level. If 110 cubes of the side of 7 cm are completely measured in it, then find the raise in water level.
(J'19)
Step 1: Calculate total volume of cubes
Side of cube = 7 cm = 0.07 m
Volume of one cube = a³ = (0.07)³ = 0.000343 m³
Volume of 110 cubes = 110 × 0.000343 = 0.03773 m³
Volume of one cube = a³ = (0.07)³ = 0.000343 m³
Volume of 110 cubes = 110 × 0.000343 = 0.03773 m³
Step 2: Calculate rise in water level
Tank radius = 7 m
Let rise in water level = h m
Volume of water displaced = πr²h = (22/7) × (7)² × h = 154h m³
This equals volume of cubes: 154h = 0.03773
h = 0.03773 / 154 ≈ 0.000245 m = 0.245 mm
Let rise in water level = h m
Volume of water displaced = πr²h = (22/7) × (7)² × h = 154h m³
This equals volume of cubes: 154h = 0.03773
h = 0.03773 / 154 ≈ 0.000245 m = 0.245 mm
Calculate Water Level Rise
∴ The water level will rise by approximately 0.245 mm.
10
The sum of the radius of base and height of a solid right circular cylinder is 37 cm. If its total surface area is 1628 cm², then find the volume of the cylinder (Use π = 22/7).
(May' 2022)
Step 1: Set up equations
Given: r + h = 37 ...(1)
Total Surface Area = 2πr(h + r) = 1628
2 × (22/7) × r × 37 = 1628
(44/7) × 37r = 1628
Total Surface Area = 2πr(h + r) = 1628
2 × (22/7) × r × 37 = 1628
(44/7) × 37r = 1628
Step 2: Solve for radius
(44/7) × 37r = 1628
44 × 37r = 1628 × 7
1628r = 11396
r = 11396 / 1628 = 7 cm
44 × 37r = 1628 × 7
1628r = 11396
r = 11396 / 1628 = 7 cm
Step 3: Find height and volume
From (1): 7 + h = 37 ⇒ h = 30 cm
Volume = πr²h = (22/7) × (7)² × 30 = (22/7) × 49 × 30 = 22 × 7 × 30 = 4620 cm³
Volume = πr²h = (22/7) × (7)² × 30 = (22/7) × 49 × 30 = 22 × 7 × 30 = 4620 cm³
Calculate Cylinder Volume
∴ The volume of the cylinder is 4620 cm³.
11
A metallic vessel is in the shape of a right circular cylinder mounted over a hemisphere. The common diameter is 42 cm and the height of the cylindrical part is 21 cm. Find the capacity of the vessel. (Take π = 22/7).
(Aug' 2022)
Step 1: Calculate dimensions
Diameter = 42 cm, so radius = 21 cm
Height of cylinder = 21 cm
Height of cylinder = 21 cm
Step 2: Calculate volume of cylinder
Volume of Cylinder = πr²h = (22/7) × (21)² × 21
= (22/7) × 441 × 21 = 22 × 63 × 21 = 29106 cm³
= (22/7) × 441 × 21 = 22 × 63 × 21 = 29106 cm³
Step 3: Calculate volume of hemisphere
Volume of Hemisphere = (2/3)πr³ = (2/3) × (22/7) × (21)³
= (2/3) × (22/7) × 9261 = (2/3) × 22 × 1323 = (2/3) × 29106 = 19404 cm³
= (2/3) × (22/7) × 9261 = (2/3) × 22 × 1323 = (2/3) × 29106 = 19404 cm³
Step 4: Calculate total capacity
Total Capacity = Volume of Cylinder + Volume of Hemisphere
= 29106 + 19404 = 48510 cm³
= 29106 + 19404 = 48510 cm³
Calculate Vessel Capacity
∴ The capacity of the vessel is 48510 cm³.
