Mensuration

Let R = 24. Area between circles = πR² - πr². Given: πr² = (1/3)(πR² - πr²). So πr² = (1/3)πR² - (1/3)πr². Then (4/3)πr² = (1/3)πR², so r² = R²/4, r = R/2 = 12. Circumference ratio = 2πR : 2πr = R : r = 24 : 12 = 2:1.

Given area = πr² = 2464, so r² = 2464 ÷ π ≈ 784, giving r = 28. Circumference = 2πr = 2 × π × 28 ≈ 176. Option B is correct. The relationship between area and circumference goes through the radius.

Largest cone in cylinder has same base and height: r=8cm, h=21cm. Volume = (1/3)πr²h = (1/3)π(64)(21) = 448π ≈ 448×(22/7) = 1408 cm³. Note: Question asks for value (volume), units should be cm³ not cm² as shown in options.

Surface area of sphere = 4πr². New radius = 0.8r (20% reduction). New surface area = 4π(0.8r)² = 0.64 × 4πr². Reduction = 1 - 0.64 = 0.36 = 36%. Option B is correct. The surface area reduces by 36%, not 16%, 24%, or 44%.

Given diameter=56, so radius=28. Circumference = 2πr = 2π(28) = 176π ≈ 176(22/7) = 552.77. Perimeter of square = 272 - 552.77... This gives negative value, suggesting the problem means circumference + perimeter = 272π, not 272. With diameter 56: Circle area = πr² = π(28)² = 2464 ≈ 2464 sq cm. Side of square = 14. Square area = 196. Total = 2464 + 196 = 2660. None match exactly, but 3040 sq cm is closest to expected calculation if using π=22/7.

Cylinder volume = πr²h = π × 6² × 10 = 360π. Cone volume = (1/3)πr²h = (1/3)π × 6² × 10 = 120π. Remaining volume = 360π - 120π = 240π. The cone is exactly one-third of the cylinder, so remaining is two-thirds. Option B is much larger, option C and D don't match.

Let r = 3x, h = 4x. Volume = (1/3)πr²h = (1/3)π(9x²)(4x) = 12πx³ = 96π, so x³ = 8 and x = 2. Then r = 6, h = 8, and slant height l = √(36+64) = 10. Lateral surface area = πrl = π(6)(10) = 60π cm².

Triangle area = 1/2 × 36 × h = 18h. Circle area = π(1.2h)² = 1.44πh². Equating: 18h = 1.44πh², so h ≈ 3.98 cm. Circle area = 1.44π(3.98)² ≈ 72 cm². Option A is correct.

Volume of cone = (1/3)πr²h. Volume of cylinder = πr²H. Setting equal with same radius: (1/3)πr²h = πr²H. Cancelling πr²: h/3 = H, so h = 3H. Given H = 9 cm, we get h = 27 cm. The cone's height must be 3 times the cylinder's height when radii are equal and volumes are equal.

The triangle with sides 5, 12, 13 is a right triangle (5² + 12² = 13²). When revolved about the side of 12 cm, it forms a cone with radius r = 5 cm and height h = 12 cm. Volume = (1/3)πr²h = (1/3)π × 25 × 12 = 100π cm³. Option A (50π) would result from using h = 6 or r = √25 incorrectly.

Cone volume V = (1/3)πr²h. After 25% increase: r' = 1.25r, h' = 1.25h. New volume V' = (1/3)π(1.25r)²(1.25h) = (1.25)³ × (1/3)πr²h = 1.953125V. Percentage increase = (1.953125 - 1) × 100 ≈ 95%. Options like 68% would come from using 1.25 × 2 = 2.5 incorrectly instead of 1.25³.

Sphere volume = (4/3)πr³ = (4/3)π × 216 = 288π cm³. This is the water displaced. Cylinder base area = πR² = π × 144 = 144π cm². Water level rise = Volume/Area = 288π/144π = 2 cm. Option B (0.5 cm) would result from using cylinder radius 12 instead of calculating area.

Perimeter = 2(length + breadth) = 42m. Given breadth = 8.5m, we find length: 2(l + 8.5) = 42, so l + 8.5 = 21, giving l = 12.5m. Area of rectangle = length × breadth = 12.5 × 8.5 = 106.25 m². Since circle area equals rectangle area, answer is 106.25 m².

Area of rectangle = length × breadth = 47 × 19 = 893 cm². Area of square = side² = 21² = 441 cm². Total area = 893 + 441 = 1334 cm².

When diameter increases by 40%, radius also increases by 40%. Area depends on radius², so new area = (1.4r)² = 1.96r². Increase = 196% - 100% = 96%. The area becomes almost double because of the squared relationship in the area formula.