Question 1

A solid sphere has a radius of 7 cm. What is the volume of the sphere? (Use π = 22/7)

Accepted Answer

Step 1: Recall the formula for volume of a sphere: V = (4/3)πr³. Step 2: Substitute r = 7 cm and π = 22/7. Step 3: V = (4/3) × (22/7) × 7³ = (4/3) × (22/7) × 343. Step 4: V = (4/3) × 22 × 49 = (4 × 22 × 49)/3 = 4312/3 ≈ 1437.33 cm³. Step 5: Simplifying: V = (4 × 22 × 343)/(3 × 7) = (4 × 22 × 49)/3 = 4312/3. Calculating: 4312 ÷ 3 = 1437⅓ cm³ or exactly 1437.33 cm³. However, for cleaner computation: (4/3) × (22/7) × 343 = (88 × 343)/21 = 30184/21 = 1437⅓ cm³. Rounding to standard form: approximately 1437.33 cm³, but the exact answer is 4312/3 cm³ or 1437⅓ cm³. For SSC purposes, this equals **1437.33 cm³** or **4312/3 cm³**. Let me recalculate cleanly: V = (4/3) × (22/7) × 343 = (4 × 22 × 343)/(3 × 7) = (4 × 22 × 49)/3 = 4312/3 cm³. In decimal: **1437.33 cm³** (approximately). Standard answer: **1437⅓ cm³** or **4312/3 cm³**. For multiple choice, this is typically presented as **1437.33 cm³** or rounded to **1437 cm³** or **1440 cm³**.

Question 2

A solid cylinder has radius 7 cm and height 20 cm. It is melted and recast into a solid cone of the same radius. What is the height of the cone?

Accepted Answer

Step 1: Volume of cylinder = πr²h = π × 7² × 20 = 980π cm³. Step 2: Volume of cone = (1/3)πr²H, where r = 7 cm (same radius). Step 3: Set volumes equal: (1/3) × π × 49 × H = 980π. Step 4: H = (980 × 3) / 49 = 2940 / 49 = 60 cm. Therefore, the height of the cone is 60 cm.

Question 3

A solid metallic sphere of radius 6 cm is melted and recast into a solid cone with base radius 4 cm. If the height of the cone is h cm, and a cylindrical hole of radius 2 cm and height h cm is drilled through the cone's axis, find the volume of the remaining solid (in cm³).

Accepted Answer

Step 1: Volume of sphere = (4/3)πr³ = (4/3)π(6)³ = 288π cm³. Step 2: This volume equals the cone before drilling: (1/3)πR²h = (1/3)π(4)²h = 288π, so 16h/3 = 288, giving h = 54 cm. Step 3: Volume of cylindrical hole = πr²h = π(2)²(54) = 216π cm³. Step 4: Remaining volume = 288π − 216π = 72π ≈ 226.19 cm³. However, for exact answer: 72π cm³ or approximately 226 cm³ (if numerical form required).

Question 4

A solid cone and a solid sphere have the same radius. The height of the cone equals the diameter of the sphere. If the volume of the cone is 72π cm³, what is the volume of the sphere (in cm³)?

Accepted Answer

Step 1: Let radius = r cm. Height of cone = 2r cm (given: height = diameter of sphere). Step 2: Volume of cone = (1/3)πr²h = (1/3)πr²(2r) = (2/3)πr³ = 72π. Step 3: Solving for r³: (2/3)r³ = 72, so r³ = 108. Step 4: Volume of sphere = (4/3)πr³ = (4/3)π(108) = 144π cm³. Alternative check: Cone volume = 72π, so r³ = 108. Sphere = (4/3) × 108π = 144π.

Question 5

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: 4 × (22/7) × r² = 616. Step 3: (88/7) × r² = 616. Step 4: r² = 616 × (7/88) = 4312/88 = 49. Step 5: r = 7 cm. Therefore, the radius is 7 cm.

Question 6

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

Accepted Answer

Step 1: Volume of cylinder = πr²h. Step 2: Substitute V = 1540 cm³, h = 10 cm, π = 22/7. Step 3: 1540 = (22/7) × r² × 10. Step 4: 1540 = (220/7) × r². Step 5: r² = 1540 × (7/220) = 10780/220 = 49. Step 6: r = 7 cm. Therefore, the answer is 7 cm.

Question 7

A sphere has a radius of 7 cm. What is its surface area (in cm²)?

Accepted Answer

Step 1: Surface Area of sphere = 4πr². Step 2: Substitute r = 7 cm. Step 3: SA = 4 × (22/7) × 7² = 4 × (22/7) × 49 = 4 × 22 × 7 = 616 cm². Therefore, the answer is 616 cm².

Question 8

A cone has a base radius of 7 cm and height of 6 cm. What is its volume? (Use π = 22/7)

Accepted Answer

Step 1: Volume of cone = (1/3)πr²h. Step 2: V = (1/3) × (22/7) × 7² × 6. Step 3: V = (1/3) × (22/7) × 49 × 6. Step 4: V = (22 × 49 × 6)/(3 × 7) = (22 × 7 × 6)/3 = 924/3 = 308 cm³. Therefore, the answer is 308 cm³.

Question 9

A cylinder has a volume of 1540 cm³ and 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 10

The volume of a cone is 616 cm³ and its height is 12 cm. What is the radius of its base? (Use π = 22/7)

Accepted Answer

Step 1: Volume of cone = (1/3)πr²h. Step 2: (1/3) × (22/7) × r² × 12 = 616. Step 3: (22/7) × r² × 4 = 616. Step 4: (88/7) × r² = 616. Step 5: r² = 616 × (7/88) = 4312/88 = 49. Step 6: r = 7 cm. Therefore, the radius is 7 cm.

Question 11

A cylinder has a total surface area of 462 cm² and radius 7 cm. What is its height?

Accepted Answer

Step 1: Total Surface Area of cylinder = 2πr(r + h). Step 2: 2 × (22/7) × 7 × (7 + h) = 462. Step 3: 2 × 22 × (7 + h) = 462. Step 4: 44(7 + h) = 462. Step 5: 7 + h = 462/44 = 10.5. Step 6: h = 10.5 - 7 = 3.5 cm. Therefore, the height is 3.5 cm.

Question 12

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 a cylinder = πr²h. Step 2: Volume of a cone = (1/3)πr²h. Step 3: For cylinder: V_cyl = π × 5² × 12 = 300π cm³. Step 4: For cone: V_cone = (1/3) × π × 5² × 12 = 100π cm³. Step 5: Ratio = V_cyl : V_cone = 300π : 100π = 3 : 1.

Question 13

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) = 4312/88 = 49. Step 6: r = 7 cm. Therefore, the answer is 7 cm.

Question 14

The volume of a sphere is 288π cm³. What is its radius?

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 answer is 6 cm.

Question 15

The volume of a sphere is 288π cm³. What is its radius?

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 16

A sphere of radius 6 cm is melted and recast into a cylinder of radius 4 cm. What is the height of the cylinder (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 cylinder = πr²h = π(4)²h = 16πh cm³. Step 3: Since the sphere is melted and recast: 288π = 16πh. Step 4: h = 288π/(16π) = 288/16 = 18 cm.

Question 17

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 = (22/7) × r × 25. Step 3: 550 = (550r)/7. Step 4: r = (550 × 7)/550 = 7 cm.

Question 18

The curved surface area of a cone is 550 cm² and its slant height is 25 cm. What is 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) = 22 × 7/22 = 7 cm (using π = 22/7). Therefore, radius = 7 cm.

Question 19

A solid cylinder has radius 7 cm and height 10 cm. A cone with the same radius and height is carved out from the top. What is the volume of the remaining solid (in cm³)?

Accepted Answer

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

Question 20

A cone and a sphere have equal volumes. The cone has radius 6 cm and height 8 cm. What is the radius of the sphere (in cm)?

Accepted Answer

Step 1: Volume of cone = (1/3)πr²h = (1/3)π(6)²(8) = (1/3)π(36)(8) = 96π cm³. Step 2: Volume of sphere = (4/3)πR³, where R is the radius of sphere. Step 3: Since volumes are equal: 96π = (4/3)πR³. Step 4: 96 = (4/3)R³. Step 5: R³ = 96 × (3/4) = 72. Step 6: R = ∛72 = ∛(8 × 9) = 2∛9 cm ≈ 4.16 cm. For exact answer: R = ∛72 = 2∛9 cm or approximately 4.2 cm (if using calculator: 72^(1/3) ≈ 4.16).

Question 21

A cone has a base radius of 5 cm and height of 12 cm. If the slant height is increased by 20% while keeping the base radius constant, what is the percentage increase in curved surface area?

Accepted Answer

Step 1: Original slant height l = √(r² + h²) = √(25 + 144) = √169 = 13 cm. Step 2: Original CSA = πrl = π(5)(13) = 65π cm². Step 3: New slant height = 13 × 1.20 = 15.6 cm. Step 4: New CSA = π(5)(15.6) = 78π cm². Step 5: Percentage increase = [(78π − 65π)/(65π)] × 100 = (13π/65π) × 100 = (13/65) × 100 = 20%.

Question 22

A solid cone has base radius 5 cm and height 12 cm. It is placed inside a cylinder of the same radius and height. What percentage of the cylinder's volume is NOT occupied by the cone?

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: Unoccupied volume = 300π − 100π = 200π cm³. Step 4: Percentage = (200π/300π) × 100 = (200/300) × 100 = (2/3) × 100 = 66.67% (or 66⅔%).

Question 23

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: Calculate volume of sphere using V = (4/3)πr³. Volume of sphere = (4/3)π(6)³ = (4/3)π(216) = 288π cm³. Step 2: The volume of cone equals volume of sphere. Using V = (1/3)πr²h for cone: 288π = (1/3)π(4)²h. Step 3: Simplify: 288π = (1/3)π(16)h → 288 = (16h)/3 → h = (288 × 3)/16 = 864/16 = 54 cm. Therefore, the height of the cone is 54 cm.

Question 24

A cylinder has a radius of 7 cm and height of 10 cm. A cone with the same radius and height is placed inside it. What is the difference between their volumes (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: Difference = 490π - (490π)/3 = (1470π - 490π)/3 = (980π)/3 = (980 × 22)/(3 × 7) = 21560/21 ≈ 1026.67 cm³. Using π = 22/7: Difference = 980 × 22 / (3 × 7) = 21560/21 = 1026.67 cm³. Alternatively: Difference = 2 × (cone volume) = 2 × (490π/3) = 980π/3 ≈ 1026.67 cm³. Rounding to nearest integer: 1027 cm³.

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