Mensuration

Volume of cylinder = πr²h. Statement II gives r² = 616/π, so r² is known. Statement III gives h = 28m. With II and III, we can calculate volume. Statement I says r = h/2, which with III gives r = 14m, then with h = 28, we can calculate volume. With I and II, from II we have r² = 616/π, and from I we have h = 2r, so we can calculate volume. Thus any two statements are sufficient.

Original CSA = 2πrh. New height = 1.2h, new radius = 0.9r. New CSA = 2π(0.9r)(1.2h) = 2πrh × 1.08. Change = 8% increase. Percentage change = (new - original)/original × 100 = 8%.

Original area = πr². New area = π(r+2)² = πr² + 44. So π(r+2)² - πr² = 44 → π(r² + 4r + 4 - r²) = 44 → π(4r + 4) = 44 → 4π(r + 1) = 44 → r + 1 = 11/π ≈ 3.5 → r ≈ 2.5 cm.

Circumference C = 2πr = 26.4 cm, so r = 26.4 ÷ (2π) = 13.2 ÷ π. Area = πr² = π × (13.2/π)² = π × 174.24/π² = 174.24/π. Using π ≈ 22/7: 174.24 × 7/22 = 55.44 cm². Option C is correct.

Circle area = 144.5 cm², so square area = 2 × 144.5 = 289 cm². Side of square = √289 = 17 cm. Sum of four sides (perimeter) = 4 × 17 = 68 cm. However, the question asks for 'sum of the sides' which could mean perimeter (68 cm), but the correct answer is 34 cm, suggesting it may mean sum of two adjacent sides or there's an interpretation issue. Given the options, 34 cm is marked correct, which equals 2 × 17.

Sphere volume = (4/3)πr³. Cone volume = (1/3)πr²h. Since volumes are equal and radii are equal: (4/3)πr³ = (1/3)πr²h. Simplifying: 4r = h, so h : r = 4 : 1.

Volume of a prism = base area × height. Base is a square with side 8 cm, so base area = 8 × 8 = 64 cm². Height = 10 cm. Volume = 64 × 10 = 640 cm³. Option A (680) might come from 68 × 10, confusing perimeter with area.

Volume of cylinder = πr²h. If volumes are equal: πr₁²h₁ = πr₂²h₂, giving r₁²h₁ = r₂²h₂. With height ratio 1:2 (h₁/h₂ = 1/2), we get r₁²/r₂² = h₂/h₁ = 2/1, so r₁/r₂ = √2. Statements I and II are sufficient. Statement III introduces unnecessary complexity about sphere area.

To find the area from circumference, first find the radius. Circumference C = 2πr, so r = C/(2π) = 132/(2π) = 21 cm. Area = πr² = π(21)² = 1386 cm². Option B is correct. Options A, C, and D are calculation errors - they may have used wrong radius values or incorrect area formulas.

When a solid is melted and recast, its volume remains constant. Volume of hemisphere = (2/3)πr³. Volume of cylinder = πR²h, where R = 9 cm (diameter/2) and h = 162 cm. Equating: (2/3)πr³ = π(9)²(162) = π(81)(162). Simplifying: (2/3)r³ = 81 × 162 = 13122. So r³ = 13122 × 3/2 = 19683. Since 27³ = 19683, the radius of hemisphere is 27 cm.

Hemisphere radius = 15 cm. Volume of hemisphere = (2/3)π(15)³ = (2/3)π(3375) = 2250π cm³. Cylinder: diameter = 5 cm, so radius = 2.5 cm, height = 6 cm. Volume of one cylinder = π(2.5)²(6) = π(6.25)(6) = 37.5π cm³. Number of bottles = 2250π / 37.5π = 2250/37.5 = 60. The liquid volume is conserved when transferring.

Let cylinder radius = R, height = R (since they're equal). Cylinder volume = πR² × R = πR³. Volume of 48 balls = 48 × (4/3)πr³ = 64πr³. Equating volumes: πR³ = 64πr³, so R³ = 64r³, giving R = 4r. Therefore r/R = 1/4, or ratio is 1:4. Option A is correct.

The correct answer is C (1 cm). Volume of sphere = (4/3)πr³ = (4/3)π(3)³ = 36π cm³. This equals the volume of water displaced in the cylinder: πR²h = π(6)²h = 36πh cm³. Equating: 36π = 36πh, so h = 1 cm. The water level rises by 1 cm.

Volume is conserved when melting and recasting. Volume of 10 spheres = 10 × (4/3)πr³ = 10 × (4/3)π(5)³ = 10 × (500/3)π = (5000/3)π. Each cone volume = (1/3)πR²h = (1/3)π(5)²(10) = (250/3)π. Number of cones = (5000/3)π ÷ (250/3)π = 5000/250 = 20. Option A (20) is correct.

The remaining solid has a cylindrical outer surface (2πrh = 24π cm²) plus a circular top (πr² = 9π cm²). The inner conical cavity has lateral surface area πrl = π × 3 × 5 = 15π cm². Total = 24π + 9π + 15π = 48π cm². Option A (45π) incorrectly omits the top surface. Option B (55π) and D (50π) are miscalculations.