Mensuration

Let cylinder height = h, so radius = h/4. Cylinder volume = π(h/4)²h = πh³/16. Sphere volume = (4/3)π(h/4)³ = πh³/48. Number of spheres = Cylinder volume / Sphere volume = (πh³/16)/(πh³/48) = 48/16 = 3. The h³ terms cancel out neatly, showing the answer is independent of the actual height.

Volume of hemisphere = (2/3)π(2.1)³ = 19.404 cm³. Volume of cone = (1/3)π(2.1)²(4) = 18.476 cm³. Total solid volume = 37.88 cm³. Cylinder volume = π(5)²(9.8) = 769.69 cm³. Water left = 769.69 - 37.88 = 731.81 cm³, which rounds to 732 cm³. This matches option B (732.166 cm³) accounting for π approximation.

Let r=3k, h=4k. Volume = (1/3)πr²h = (1/3)π(9k²)(4k) = 12πk³ = 96π, so k³=8, k=2. Thus r=6 cm, h=8 cm. Lateral surface area = πrl, where l = sqrt(r²+h²) = sqrt(36+64) = 10 cm. LSA = π(6)(10) = 60π cm².

Area of square = 196 cm², so side = √196 = 14 cm. Radius = 14 cm. Diameter = 2 × radius = 2 × 14 = 28 cm. The key is finding the side length from the area first, then using the relationship between radius and diameter.

Given radius:height = 3:4 and height = 24 cm, so radius = 18 cm. Volume of cone = (1/3)πr²h = (1/3) × π × 18² × 24 = (1/3) × π × 324 × 24 = π × 324 × 8 = 2592π ≈ 8143 cm³. Since 2592 cm³ is NOT the volume (that would be without π), and all specific options A-D are incorrect, the correct answer is E. Option D (2592) would be correct only if π = 1, which is false.

Curved surface area of cone = πrl where r is radius and l is slant height. Given CSA = 355 cm² and r = 11.3 cm. Therefore l = 355/(π×11.3) = 355/35.5 = 10 cm (using π ≈ 3.14, π×11.3 ≈ 35.5).

For a square prism with base side x and height 10, surface area = 2x² + 4(x)(10) = 2x² + 40x = 192. This gives x² + 20x - 96 = 0, so (x + 24)(x - 4) = 0, hence x = 4. Volume = base area × height = x² × 10 = 16 × 10 = 160 cm³.

Surface area of cube = 6a² where a is side length. If 6a² = 216, then a² = 36, so a = 6 m. Volume = a³ = 6³ = 216 m³.

To find area, we need both length and breadth. Statement A gives perimeter: 2(l+b) = 80. Statement C gives breadth = 15. With A and C together, we can find l = (80/2) - 15 = 25, then area = 25 × 15 = 375. Statement B gives cost per square meter but not total cost, so it's useless alone. So A+C together are sufficient, but this exact combination isn't listed in D (which says A+B, B+C, or C+A - all with B). Since the correct pair A+C isn't listed, E is correct.

Volume of sphere = (4/3)πr³ = (4/3)π(3)³ = 36π cm³. Wire is a cylinder with diameter 0.2 cm, so radius = 0.1 cm. Volume of cylinder = πr²h = π(0.1)²h = 0.01πh. Equating: 36π = 0.01πh, so h = 3600 cm = 36 m. Option C is correct.

For a cylinder with radius r and height h: curved surface area = 2πrh, and area of each base = πr². Given 4(2πrh) = 6(2πr²), we get 8πrh = 12πr², so 2h = 3r, giving r = 8 when h = 12. Volume = πr²h = π(64)(12) = 768π.

The volume of the spherical ball equals the volume of displaced water. Volume of water raised = π×12^2×6.75 = π×144×6.75 = π×972. Equating to (4/3)πr^3 gives r^3 = 729, so r = 9 cm. Option C is correct.

For a cylinder, curved surface area = 2πrh = 264 and volume = πr²h = 924. Dividing volume by CSA: (πr²h)/(2πrh) = 924/264, which simplifies to r/2 = 3.5, giving r = 7. Substituting r in CSA: 2π(7)h = 264, so h = 264/(14π) = 18.857/π. Since π cancels in the ratio, diameter:height = 14:(18.857/π) = 14π:18.857. Simplifying gives 7:3, confirming option B. The mixed units (m and cm) don't affect the ratio.

Volume of cone = (1/3)πr²h. Given r = 3.5 cm = 7/2 cm, h = 12 cm. V = (1/3) × (22/7) × (7/2)² × 12 = (1/3) × (22/7) × (49/4) × 12 = 22 × 7 = 154 cm³. Option A (308) would be without the 1/3 factor, option C (77) uses 6 instead of 12, and option D (231) is miscalculated.

When a solid is melted and recast, its volume is conserved. Cube volume = 10³ = 1000 cm³. Cone volume = (1/3)πr²h = (1/3)π × 7² × h = (49π/3)h. Equating: (49π/3)h = 1000, so h = 3000/(49π) ≈ 19.48 cm.