Mensuration

Volume of large sphere = (4/3)π(8)³ = (4/3)π × 512. Volume of each small ball = (4/3)π(1)³ = (4/3)π. Number of balls = (4/3)π × 512 / ((4/3)π) = 512. Option A (256) would give radius ≈ 6.3, C (768) requires larger sphere, D (1024) needs radius ≈ 10.

Surface area of sphere = 4πr² = 616, so r² = 616/(4π) = 154/π ≈ 49, giving r ≈ 7 cm. If area reduces by 75%, new area = 25% of 616 = 154. Then 4πR² = 154, so R² ≈ 12.25, R ≈ 3.5 cm. The answer matches option D.

For cone: V = (1/3)πr²h = 1232, h = 24. So (1/3)πr² × 24 = 1232, giving r² = 1232 × 3/(24π) = 49, thus r = 7 cm. Curved surface area = πrl = πr√(r² + h²) = π × 7 × √(49 + 576) = π × 7 × 25 ≈ 550 cm². Options B (704), C (924), D (1254) are incorrect - likely using wrong formulas or calculation errors.

CSA = 2πrh, TSA = 2πr(r+h). Given: CSA = TSA/3 → 2πrh = 2πr(r+h)/3 → 6πrh = 2πr(r+h) → 6h = 2(r+h) → 6h = 2r + 2h → 4h = 2r → r = 2h = 2(9) = 18 m. Check: CSA = 2π(18)(9) = 324π, TSA = 2π(18)(27) = 972π, CSA/TSA = 324/972 = 1/3 ✓. Option A (9 m), C (27 m), and D (30 m) don't satisfy the condition.

Square perimeter = 8m, so each side = 8 ÷ 4 = 2m. This is the circle's radius. Area of circle = πr² = π × 2² = 4π. Using π ≈ 22/7: 4 × 22/7 = 88/7 ≈ 12.57 m². But wait, the options are much larger (1156-1486). Let me recalculate: if the area is 1386 m², then r² = 1386/π = 1386/(22/7) = 1386 × 7/22 = 441. So r = √441 = 21m. For a square with side 21m, perimeter = 84m, not 8m. There's a unit error in the question or options. Assuming the question means perimeter = 88m (not 8m), then side = 22m, radius = 22m, area = π × 22² = (22/7) × 484 = 22 × 69.14 ≈ 1521. None of the options match exactly. However, if we consider 1386 m² with π=22/7: r² = 1386 × 7/22 = 441, r = 21m. This would mean square side = 21m, perimeter = 84m. The question likely has a typo (8 instead of 84), but 1386 m² is the intended answer based on standard values.

The largest circle inside a rectangle has a diameter equal to the shorter side. Here, the rectangle is 21 cm by 30 cm, so the maximum diameter is 21 cm, making the radius 10.5 cm. Area = pi * r^2 = (22/7) * 10.5 * 10.5 = 346.5 cm^2. Option A (338.8) is incorrect calculation. Option C (358.2) uses wrong dimensions. Option D (340.6) is a miscalculation.

Cylinder volume = πr²h = π × 3² × 5 = 45π cm³. Cone volume = (1/3)πr²h. Note: cone radius is 1 mm = 0.1 cm. Cone volume = (1/3) × π × (0.1)² × 1 = π/300 cm³. Number of cones = Cylinder volume / Cone volume = 45π / (π/300) = 45 × 300 = 13500.

Circle diameter = 70 cm, so radius = 35 cm. Area of circle = πr² = (22/7) × 35² = 22 × 5 × 35 = 3850 cm². Total area = 5450 cm², so square area = 5450 - 3850 = 1600 cm². Square side = √1600 = 40 cm. Circumference = 2πr = 2 × (22/7) × 35 = 220 cm. Perimeter of square = 4 × 40 = 160 cm. Difference = 220 - 160 = 60 cm. None of the options match 60 cm, so E is correct.

Volume of cylinder = π(2)²(8) = 32π m³. This equals the cone's volume: (1/3)π(1.5)²h = (1/3)π(2.25)h = 0.75πh. Equating: 0.75πh = 32π, so h = 32/0.75 = 42.67 m. The other options don't satisfy the volume conservation equation.

Original area = πr². New area = π(r+1)² = π(r² + 2r + 1). Increase = π(2r + 1) = 22. So 2r + 1 = 22/π ≈ 7, giving r ≈ 3. Using π ≈ 22/7: 2r + 1 = 22/(22/7) = 7, so 2r = 6, r = 3 cm. Checking: Original area = 9π ≈ 28.3, new area = 16π ≈ 50.3, difference = 7π ≈ 22 ✓

Metal volume = π(R² - r²)h = 748. With R=9, h=14: π(81 - r²)14 = 748 → (81 - r²) = 748/(14π) ≈ 17 → r² ≈ 64 → r ≈ 8. So thickness = R - r = 9 - 8 = 1 cm. Option B (2.3) is too large, C and D are even larger.

Circle area = πr² with r = 14 cm (half of diameter 28). Area = 616 cm². Rectangle area = 1166 - 616 = 550 cm². With length 25 cm, width = 550/25 = 22 cm. Perimeter = 2(25+22) = 94 cm. Circumference = 2πr = 88 cm. Sum = 88 + 94 = 182 cm.

Area of square = 196 cm², so side = √196 = 14 cm. This equals the radius of the circle. Diameter = 2 × radius = 28 cm. Length of rectangle = 2 × diameter = 56 cm. Breadth = length/2 = 28 cm. Area of rectangle = length × breadth = 56 × 28 = 1568 cm². Option C is correct.

Volume of the sphere = (4/3)πr³ = (4/3)π(9)³ = 972π cm³. The wire is a cylinder with radius 0.2 cm (half of diameter 0.4 cm). Volume of wire = πr²h = π(0.2)²h. Setting equal: 972π = 0.04πh, giving h = 24300 cm = 243 m.

Area of circle = πr², diameter = 2r. Given πr² : 2r = 22 : 1, so πr/2 = 22, meaning r = 44/π. Using π ≈ 22/7: r = 44/(22/7) = 44 × 7/22 = 14 units.