Mensuration

Copper wire is cylindrical with r = 1 mm, h = 36 m = 36000 mm. Volume = π(1)²(36000) = 36000π mm³. For sphere with same volume: (4/3)πR³ = 36000π, giving R³ = 27000, so R = 30 mm = 3 cm.

Pipe: r = 7 cm, flow rate = 5 m/s = 500 cm/s. Cross-section area = π(7)² = 49π cm². Volume flow rate = 49π × 500 = 24500π cm³/s. Tank: base = 3×5 = 15 m² = 150000 cm², height = 1.54 m = 154 cm. Tank volume = 150000 × 154 = 23,100,000 cm³. Time = 23,100,000/(24500π) ≈ 300 seconds = 5 minutes.

Curved surface area = πrl, where l = √(r² + h²). Given r:h = 3:4, we can write r = 3k, h = 4k, but without any actual value (radius, height, or slant height), the area cannot be calculated numerically.

Volume of hemispherical bowl = (2/3) × π × 20³ = 16,000π/3 cm³. Volume of each conical bottle = (1/3) × π × 5² × 8 = 200π/3 cm³. Number of bottles = (16,000π/3) ÷ (200π/3) = 16,000 ÷ 200 = 80 bottles.

Volume of each cube = 216 m³, so side = ³√216 = 6 m. Three cubes joined end-to-end form a cuboid of dimensions 6m × 6m × 18m. Surface area = 2(lb + bh + hl) = 2(6×6 + 6×18 + 18×6) = 2(36 + 108 + 108) = 2(252) = 504 m². Option D is correct.

For a triangle, Area = (1/2) × Perimeter × In-radius. Area = (1/2) × 32 × 4 = 64 cm². Volume of prism = Base Area × Height. 376 = 64 × Height, so Height = 376/64 = 5.875 cm.

Original circle area = π × 10² = 100π. 40% removed, so remaining area = 60% of 100π = 60π. This becomes the slant surface area of the cone (curved surface area = π × r × l, where r = l here).

For a circle of radius r, circumference = 2πr and area = πr². When r = 6 cm, circumference = 12π and area = 36π. The ratio 12π:36π simplifies to 1:3 by dividing both terms by 12π. Option B (2:3) is incorrect because it doesn't account for the π in both terms cancelling out.

Square perimeter = 56, so side = 56/4 = 14 m. This side equals the circle's radius (r = 14). Area of circle = πr² = (22/7) × 14² = (22/7) × 196 = 22 × 28 = 616 sqm. Option C (606) and B (566) are calculation errors. Option D has inconsistent units (sq instead of sqm).

Let sphere radius = r and cube side = a. Surface area of sphere = 4πr², cube = 6a². Since equal: 4πr² = 6a², so r² = 3a²/(2π) and r = a√(3/(2π)). Volume of sphere = (4/3)πr³ = (4/3)πa³(3/(2π))^∕², cube = a³. Ratio of squares of volumes = [(4/3)πa³(3/(2π))^∕²]² : (a³)² = (16π/9)a⁶(3/(2π)) : a⁶ = (16π/9)(3/(2π)) = 8/3 = (6)/π.

Let edge of cube be 'a'. Volume = a³. Sum of edges = 12a (12 edges each of length a). Given a³ = 12a, so a² = 12 (a ≠ 0). Total surface area = 6a² = 6 × 12 = 72 square units. Option D is correct.

Volume is conserved when melting and reshaping. Cylinder volume = π × 3² × 5 = 45π cm³. Converting to mm³: 45π × 1000 = 45000π mm³. Each cone volume = (1/3) × π × (1)² × 1 = π/3 mm³. Number of cones = 45000π ÷ (π/3) = 45000 × 3 = 135000. The question likely means 1 mm radius cones (not 1 cm). Options A, B, C are incorrect.

Large hemisphere: R=7.5m, volume=(2/3)π(7.5)³, surface area=3π(7.5)²=168.75π m² (curved+base). Small hemispheres: r=1.5m, volume each=(2/3)π(1.5)³. Number=Volume large/Volume small=(7.5/1.5)³=125. Total SA of small hemispheres=125×3π(1.5)²=843.75π m². Increase=843.75π-168.75π=675π. Percentage=675π/168.75π×100=400%.

Curved surface area of cylinder = 2πrh. Given CSA = 4πrh. So 2πrh = 4πrh, which is contradictory unless we recognize that the given CSA formula has a coefficient of 4, meaning h is actually 2h in the formula. If CSA = 4πrh and the standard formula is 2πrH, then H = 2h.

Hemisphere volume = (2/3)πr³ where r = 27 cm, giving 13122π cm³. Each cylinder has volume πr²h = π × 9 × 9 = 81π cm³. Number of bottles = 13122π ÷ 81π = 162. Option A (81) is exactly half the correct answer - likely forgetting the 2/3 factor in hemisphere formula. Option C (324) is double the correct answer.