10th Maths Mensuration Exercise 10.2 Solutions

Exercise 10.2 Solutions – Class X Mathematics

Exercise 10.2 Solutions

Class X Mathematics textbook by the State Council of Educational Research and Training, Telangana, Hyderabad

Problem 1

A toy is in the form of a cone mounted on a hemisphere of the same diameter. The diameter of the base and the height of the cone are 6 cm and 4 cm respectively. Determine the surface area of the toy. [use π = 3.14]

[Diagram description: Cone with height 4cm mounted on hemisphere with diameter 6cm]

Solution:

Given: Diameter = 6 cm ⇒ Radius (r) = 3 cm

Cone height (h) = 4 cm

Slant height of cone (l) = √(r² + h²) = √(9 + 16) = 5 cm

Curved surface area of cone = πrl = 3.14 × 3 × 5 = 47.1 cm²

Curved surface area of hemisphere = 2πr² = 2 × 3.14 × 9 = 56.52 cm²

Total surface area = Cone CSA + Hemisphere CSA = 47.1 + 56.52 = 103.62 cm²

Problem 2

A solid is in the form of a right circular cylinder with a hemisphere at one end and a cone at the other end. The radius of the common base is 8 cm and the heights of the cylindrical and conical portions are 10 cm and 6 cm respectively. Find the total surface area of the solid. [use π = 3.14]

[Diagram description: Cylinder (height 10cm) with hemisphere on one end and cone (height 6cm) on other end, all with radius 8cm]

Solution:

Given: Radius (r) = 8 cm

Cylinder height (h₁) = 10 cm, Cone height (h₂) = 6 cm

Cylinder CSA = 2πrh₁ = 2 × 3.14 × 8 × 10 = 502.4 cm²

Cone slant height (l) = √(r² + h₂²) = √(64 + 36) = 10 cm

Cone CSA = πrl = 3.14 × 8 × 10 = 251.2 cm²

Hemisphere CSA = 2πr² = 2 × 3.14 × 64 = 401.92 cm²

Total surface area = Cylinder CSA + Cone CSA + Hemisphere CSA = 502.4 + 251.2 + 401.92 = 1155.52 cm²

Problem 3

A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the capsule is 14 mm and the thickness is 5 mm. Find its surface area.

[Diagram description: Cylinder with two hemispheres on both ends, total length 14mm, diameter 5mm]

Solution:

Given: Total length = 14 mm, Diameter = 5 mm ⇒ Radius (r) = 2.5 mm

Height of cylinder = Total length – 2 × radius = 14 – 5 = 9 mm

Cylinder CSA = 2πrh = 2 × 3.14 × 2.5 × 9 ≈ 141.3 mm²

Two hemispheres = 1 full sphere surface area = 4πr² = 4 × 3.14 × 6.25 ≈ 78.5 mm²

Total surface area = 141.3 + 78.5 ≈ 219.8 mm²

Problem 4

Two cubes each of volume 64 cm³ are joined end to end together. Find the surface area of the resulting cuboid.

Solution:

Volume of each cube = 64 cm³ ⇒ Side length (a) = ∛64 = 4 cm

When joined, cuboid dimensions become: Length = 8 cm, Breadth = 4 cm, Height = 4 cm

Total surface area = 2(lb + bh + hl) = 2(8×4 + 4×4 + 4×8) = 2(32 + 16 + 32) = 160 cm²

Problem 5

A storage tank consists of a circular cylinder with a hemisphere stuck on either end. If the external diameter of the cylinder be 1.4 m and its length be 8 m, find the cost of painting it on the outside at rate of ₹20 per m².

[Diagram description: Cylinder (length 8m) with two hemispheres on both ends, diameter 1.4m]

Solution:

Given: Diameter = 1.4 m ⇒ Radius (r) = 0.7 m, Cylinder height (h) = 8 m

Cylinder CSA = 2πrh = 2 × (22/7) × 0.7 × 8 = 35.2 m²

Two hemispheres = 1 full sphere surface area = 4πr² = 4 × (22/7) × 0.49 ≈ 6.16 m²

Total surface area = 35.2 + 6.16 = 41.36 m²

Cost of painting = 41.36 × 20 = ₹827.20

Problem 6

A sphere, a cylinder and a cone have the same radius and same height. Find the ratio of their volumes.

[Diagram description: Sphere, cylinder, and cone with same radius and height]

Solution:

Given: Same radius (r) and height (h), and for sphere: diameter = height ⇒ h = 2r

Volume of sphere = (4/3)πr³

Volume of cylinder = πr²h = πr²(2r) = 2πr³

Volume of cone = (1/3)πr²h = (1/3)πr²(2r) = (2/3)πr³

Ratio = Sphere : Cylinder : Cone = (4/3) : 2 : (2/3) = 4 : 6 : 2 = 2 : 3 : 1

Problem 7

A hemisphere is cut out from one face of a cubical wooden block such that the diameter of the hemisphere is equal to the side of the cube. Determine the total surface area of the remaining solid.

[Diagram description: Cube with hemisphere removed from one face, diameter of hemisphere equals side of cube]

Solution:

Let side of cube = a ⇒ Radius of hemisphere = a/2

Total surface area of cube = 6a²

Area removed (circle) = π(a/2)² = πa²/4

Curved surface area added by hemisphere = 2π(a/2)² = πa²/2

Net change = -πa²/4 + πa²/2 = +πa²/4

Total surface area = 6a² + πa²/4 = a²(6 + π/4)

Problem 8

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in the figure. If the height of the cylinder is 10 cm and its radius of the base is 3.5 cm, find the total surface area of the article.

[Diagram description: Cylinder (height 10cm, radius 3.5cm) with hemispheres scooped out from both ends]

Solution:

Given: Radius (r) = 3.5 cm, Cylinder height = 10 cm

Cylinder CSA = 2πrh = 2 × (22/7) × 3.5 × 10 = 220 cm²

Two hemispheres = 1 full sphere surface area = 4πr² = 4 × (22/7) × 12.25 = 154 cm²

Area of two circular tops removed = 2 × πr² = 2 × (22/7) × 12.25 = 77 cm²

Net surface area = Cylinder CSA + Sphere CSA – Removed circular areas = 220 + 154 – 77 = 297 cm²

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