Question 1

A cylinder has a volume of 1540 cm³ and a height of 10 cm. What is its radius? (Use π = 22/7)

Accepted Answer

Step 1: Volume of cylinder = πr²h. Step 2: 1540 = (22/7) × r² × 10. Step 3: 1540 = (220/7) × r². Step 4: r² = (1540 × 7)/220 = 10780/220 = 49. Step 5: r = 7 cm. Therefore, the answer is 7 cm.

Question 2

A cone has a base radius of 8 cm and height of 15 cm. What is its slant height?

Accepted Answer

Step 1: Slant height l is related to radius r and height h by: l² = r² + h². Step 2: Substitute r = 8 cm, h = 15 cm: l² = 8² + 15² = 64 + 225 = 289. Step 3: l = √289 = 17 cm. Therefore, the slant height is 17 cm.

Question 3

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

Accepted Answer

Step 1: Volume of sphere = (4/3)πr³. Step 2: Given volume = 288π cm³. Step 3: (4/3)πr³ = 288π. Step 4: (4/3)r³ = 288. Step 5: r³ = 288 × (3/4) = 216. Step 6: r = ∛216 = 6 cm. Therefore, the radius is 6 cm.

Question 4

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

Accepted Answer

Step 1: Surface area of sphere = 4πr². Step 2: Given SA = 616 cm². Step 3: 4πr² = 616. Step 4: 4 × (22/7) × r² = 616. Step 5: (88/7) × r² = 616. Step 6: r² = 616 × (7/88) = 4312/88 = 49. Step 7: r = 7 cm. Therefore, the radius is 7 cm.

Question 5

A cylinder and a cone have the same radius of 5 cm and the same height of 12 cm. What is the ratio of their volumes?

Accepted Answer

Step 1: Volume of cylinder = πr²h = π × 5² × 12 = 300π cm³. Step 2: Volume of cone = (1/3)πr²h = (1/3) × π × 5² × 12 = 100π cm³. Step 3: Ratio = 300π : 100π = 3 : 1. Therefore, the ratio is 3:1.

Question 6

A sphere has a radius of 7 cm. What is its surface area? (Use π = 22/7)

Accepted Answer

Step 1: Surface Area of sphere = 4πr². Step 2: SA = 4 × (22/7) × 7² = 4 × (22/7) × 49 = 4 × 22 × 7 = 616 cm². Therefore, the answer is 616 cm².

Question 7

A sphere has a surface area of 616 cm². What is its volume? [Use π = 22/7]

Accepted Answer

Step 1: Surface area of sphere = 4πr². So 4πr² = 616. Step 2: 4 × (22/7) × r² = 616. Step 3: (88/7) × r² = 616. Step 4: r² = 616 × (7/88) = 49. Step 5: r = 7 cm. Step 6: Volume = (4/3)πr³ = (4/3) × (22/7) × 343 = (4/3) × 22 × 49 = (4 × 22 × 49)/3 = 4312/3 ≈ 1437.33 cm³.

Question 8

The curved surface area of a cone is 550 cm² and its slant height is 25 cm. What is the radius of the base? [Use π = 22/7]

Accepted Answer

Step 1: Curved surface area of cone = πrl, where r is radius and l is slant height. Step 2: 550 = (22/7) × r × 25. Step 3: 550 = (550/7) × r. Step 4: r = 550 × (7/550) = 7 cm.

Question 9

The curved surface area of a cone is 550 cm² and its slant height is 25 cm. Find the radius of the base of the cone.

Accepted Answer

Step 1: Curved surface area of cone = πrl, where r is radius and l is slant height. Step 2: 550 = πrl = π × r × 25. Step 3: r = 550/(25π) = 22/(π). Step 4: Using π = 22/7: r = 22 ÷ (22/7) = 22 × (7/22) = 7 cm.

Question 10

A cylindrical tank has radius 5 m and height 12 m. It is filled with water up to a height of 9 m. A solid cone with radius 5 m and height 6 m is completely submerged in the tank. By how much does the water level rise (in metres)?

Accepted Answer

Step 1: Volume of water displaced by cone = (1/3)πr²h = (1/3)π(5)²(6) = (1/3)π(150) = 50π m³. Step 2: Cross-sectional area of cylinder = πr² = π(5)² = 25π m². Step 3: Rise in water level = Volume displaced / Cross-sectional area = 50π / 25π = 2 m. Step 4: New water level = 9 + 2 = 11 m (which is still within the tank's 12 m height).

Question 11

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 12

A solid sphere of radius 6 cm is melted and recast into a cone with base radius 6 cm. What is the height of the cone?

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, where R = 6 cm and h is unknown. Step 3: (1/3)π(6)²h = 288π. Step 4: (1/3)π(36)h = 288π. Step 5: 12πh = 288π. Step 6: h = 288/12 = 24 cm. Therefore, the answer is 24 cm.

Question 13

A cylinder and a cone have the same base radius of 5 cm and the same height of 12 cm. What is the ratio of their volumes?

Accepted Answer

Step 1: Volume of cylinder = πr²h = π(5)²(12) = 300π cm³. Step 2: Volume of cone = (1/3)πr²h = (1/3)π(5)²(12) = 100π cm³. Step 3: Ratio = 300π : 100π = 3 : 1. Therefore, the answer is 3:1.

Question 14

A cone has a base radius of 6 cm and slant height of 10 cm. What is the curved surface area of the cone?

Accepted Answer

Step 1: Curved surface area of cone = πrl, where r = radius and l = slant height. Step 2: CSA = π × 6 × 10 = 60π cm². Step 3: Using π ≈ 22/7, CSA = 60 × 22/7 ≈ 1320/7 ≈ 188.57 cm². Using π ≈ 3.14, CSA ≈ 188.4 cm². Therefore, the answer is 60π cm² or approximately 188.4 cm².

Question 15

A cylinder has a radius of 7 cm and height of 10 cm. If the radius is increased by 50% and height remains the same, by what percentage does the volume increase?

Accepted Answer

Step 1: Original volume = πr²h = π(7)²(10) = 490π cm³. Step 2: New radius = 7 × 1.5 = 10.5 cm. Step 3: New volume = π(10.5)²(10) = 1102.5π cm³. Step 4: Percentage increase = [(1102.5π - 490π) / 490π] × 100 = (612.5 / 490) × 100 = 125%. Therefore, the answer is 125%.

Question 16

A sphere has a radius of 6 cm. If the radius is increased by 50%, what is the percentage increase in the volume of the sphere?

Accepted Answer

Step 1: Original radius r₁ = 6 cm. Original volume V₁ = (4/3)πr₁³ = (4/3)π(6)³ = (4/3)π(216) = 288π cm³. Step 2: New radius r₂ = 6 + 50% of 6 = 6 + 3 = 9 cm. Step 3: New volume V₂ = (4/3)πr₂³ = (4/3)π(9)³ = (4/3)π(729) = 972π cm³. Step 4: Percentage increase = [(V₂ − V₁)/V₁] × 100 = [(972π − 288π)/(288π)] × 100 = (684π/288π) × 100 = (684/288) × 100 = 2.375 × 100 = 237.5%.

Question 17

A solid cylinder has radius 7 cm and height 10 cm. A cone with the same base radius and height is carved out from it. 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 cm³. Step 4: Using π ≈ 22/7: (980 × 22)/(3 × 7) = 21560/21 ≈ 1026.67 cm³. Or exactly: (980π)/3 ≈ 1026.67 cm³.

Question 18

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 → 288 = (16h)/3 → h = (288 × 3)/16 = 864/16 = 54 cm.

Question 19

A cylinder has radius 5 cm and height 12 cm. A cone with the same base radius and height is removed from the top. What is the remaining volume (in cm³)?

Accepted Answer

Step 1: Volume of cylinder = πr²h = π(5)²(12) = 300π cm³. Step 2: Volume of cone removed = (1/3)πr²h = (1/3)π(5)²(12) = 100π cm³. Step 3: Remaining volume = 300π − 100π = 200π = 200 × 3.14159... ≈ 628.32 cm³. For exact answer: 200π cm³ or approximately 628 cm³.

Question 20

Two spheres have radii in the ratio 2:3. If the surface area of the smaller sphere is 144π cm², find the surface area of the larger sphere (in cm²).

Accepted Answer

Step 1: Let radii be 2k and 3k. Surface area of sphere = 4πr². Step 2: For smaller sphere: 4π(2k)² = 144π → 16πk² = 144π → k² = 9 → k = 3. Step 3: Radius of smaller sphere = 2(3) = 6 cm; radius of larger sphere = 3(3) = 9 cm. Step 4: Surface area of larger sphere = 4π(9)² = 4π(81) = 324π cm².

RRB NTPC Cylinder, Cone, Sphere

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