Question 1

A rectangular prism has length 12 cm, width 8 cm, and height 5 cm. What is its total surface area?

Accepted Answer

Step 1: Surface area of rectangular prism = 2(lw + lh + wh). Step 2: = 2(12×8 + 12×5 + 8×5) = 2(96 + 60 + 40) = 2(196) = 392 cm². Therefore, answer is 392 cm².

Question 2

A cube has a side length of 6 cm. What is the total surface area of the cube?

Accepted Answer

Step 1: Surface area of cube = 6a², where a is the side length. Step 2: SA = 6 × 6² = 6 × 36 = 216 cm². Therefore, answer is 216 cm².

Question 3

The volume of a sphere is 288π cm³. What is its radius? (Use π = 22/7 or work with π symbolically)

Accepted Answer

Step 1: Volume of sphere = (4/3)πr³. Step 2: (4/3)πr³ = 288π. Step 3: (4/3)r³ = 288. Step 4: r³ = 288 × (3/4) = 216. Step 5: r = ∛216 = 6 cm. Therefore, the radius is 6 cm.

Question 4

A cylindrical tank has a radius of 5 m and height of 14 m. How many litres of water can it hold? (Use π = 22/7; 1 m³ = 1000 litres)

Accepted Answer

Step 1: Volume of cylinder = πr²h. Step 2: V = (22/7) × 5² × 14 = (22/7) × 25 × 14. Step 3: V = 22 × 25 × 2 = 1100 m³. Step 4: Convert to litres: 1100 × 1000 = 1,100,000 litres. Therefore, the answer is 1,100,000 litres.

Question 5

The total surface area of a sphere is 616 cm². What is its radius? (Use π = 22/7)

Accepted Answer

Step 1: Total Surface Area of sphere = 4πr². Step 2: 4πr² = 616. Step 3: 4 × (22/7) × r² = 616. Step 4: (88/7) × r² = 616. Step 5: r² = 616 × (7/88) = 49. Step 6: r = 7 cm. Therefore, the radius is 7 cm.

Question 6

A solid cylinder has radius 7 cm and height 10 cm. If a cone with the same radius and height is removed from the top, what is the volume of the remaining solid (in cm³)?

Accepted Answer

Step 1: Volume of cylinder = πr²h = π × 7² × 10 = 490π cm³. Step 2: Volume of cone = (1/3)πr²h = (1/3) × π × 7² × 10 = (490π)/3 cm³. Step 3: Remaining volume = 490π − (490π)/3 = (1470π − 490π)/3 = (980π)/3 = 980π/3 ≈ 1026.67 cm³. Using π ≈ 22/7: (980 × 22)/(3 × 7) = 21560/21 ≈ 1026.67 cm³.

Question 7

A sphere has a radius of 6 cm. What is the ratio of the surface area of the sphere to its volume?

Accepted Answer

Step 1: Surface area of sphere = 4πr² = 4π(6)² = 144π cm². Step 2: Volume of sphere = (4/3)πr³ = (4/3)π(6)³ = (4/3)π(216) = 288π cm³. Step 3: Ratio = 144π : 288π = 144 : 288 = 1 : 2.

Question 8

A cone is formed by rolling a sector of a circle with radius 10 cm and sector angle 216°. What is the radius of the base of the cone?

Accepted Answer

Step 1: Arc length of sector = (θ/360°) × 2πR = (216/360) × 2π × 10 = (3/5) × 20π = 12π cm. Step 2: This arc becomes the circumference of the cone's base: 2πr = 12π. Step 3: r = 6 cm.

Question 9

Two spheres have radii in the ratio 2:3. What is the ratio of their volumes?

Accepted Answer

Step 1: Volume of sphere = (4/3)πr³. Step 2: If radii are in ratio 2:3, let r₁ = 2k and r₂ = 3k. Step 3: V₁ = (4/3)π(2k)³ = (4/3)π(8k³) = (32/3)πk³. Step 4: V₂ = (4/3)π(3k)³ = (4/3)π(27k³) = (36π)k³. Step 5: Ratio V₁:V₂ = (32/3)πk³ : 36πk³ = 32 : 108 = 8 : 27.

Question 10

A solid metallic sphere of radius 6 cm is melted and recast into a solid cone with base radius 4 cm. Find the height of the cone (in cm).

Accepted Answer

Step 1: Volume of sphere = (4/3)πr³ = (4/3)π(6)³ = (4/3)π(216) = 288π cm³. Step 2: Volume of cone = (1/3)πr²h. Since volumes are equal: 288π = (1/3)π(4)²h. Step 3: 288π = (1/3)π(16)h. Step 4: 288 = (16h)/3. Step 5: h = (288 × 3)/16 = 864/16 = 54 cm.

Question 11

A cone has base radius 5 cm and slant height 13 cm. If the cone is cut by a plane parallel to the base at a height of 6 cm from the apex, what is the volume of the smaller cone (top part) that is removed (in cm³)?

Accepted Answer

Step 1: Find height of original cone using Pythagoras: h² + r² = l² → h² + 25 = 169 → h² = 144 → h = 12 cm. Step 2: The smaller cone has height 6 cm (from apex). Step 3: By similar triangles, radius of smaller cone = (6/12) × 5 = 2.5 cm. Step 4: Volume of smaller cone = (1/3)π(2.5)²(6) = (1/3)π(6.25)(6) = (1/3)π(37.5) = 12.5π ≈ 39.27 cm³.

Question 12

A cylindrical tank has radius 7 m and height 10 m. Water fills it to a height of 8 m. A solid sphere of radius 3 m is completely submerged in the water. By how much does the water level rise (in metres)?

Accepted Answer

Step 1: Volume of sphere = (4/3)π(3)³ = (4/3)π(27) = 36π m³. Step 2: Cross-sectional area of cylinder = πr² = π(7)² = 49π m². Step 3: Rise in water level = Volume of sphere / Cross-sectional area = 36π / 49π = 36/49 m ≈ 0.735 m.

Question 13

Two cones have the same height of 12 cm. The first cone has base radius 4 cm and the second has base radius 6 cm. If both cones are melted and recast into a single sphere, what is the radius of the sphere (in cm)?

Accepted Answer

Step 1: Volume of first cone = (1/3)π(4)²(12) = (1/3)π(16)(12) = 64π cm³. Step 2: Volume of second cone = (1/3)π(6)²(12) = (1/3)π(36)(12) = 144π cm³. Step 3: Total volume = 64π + 144π = 208π cm³. Step 4: Volume of sphere = (4/3)πR³ = 208π. Step 5: (4/3)R³ = 208 → R³ = (208 × 3)/4 = 156 → R = ∛156 ≈ 5.39 cm.

Question 14

A cone with base radius 8 cm and height 15 cm is placed upside down in a cylindrical container of radius 10 cm and height 20 cm. The container is filled with water up to the brim. What is the volume of water in the container (in cm³)?

Accepted Answer

Step 1: Volume of cylinder = πr²h = π(10)²(20) = 2000π cm³. Step 2: Volume of cone = (1/3)πr²h = (1/3)π(8)²(15) = (1/3)π(64)(15) = 320π cm³. Step 3: Since the cone is placed upside down (apex down) inside the cylinder and water fills to the brim, the water volume = Cylinder volume - Cone volume = 2000π - 320π = 1680π ≈ 5278.32 cm³.

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