Mensuration

The volume of the cylinder is V = πr²h = π(5)²(5) = 125π cubic cm. For a cube with edge length x, the volume is V = x³. Setting the volumes equal: x³ = 125π, which gives x = (125π)^(1/3) = 5π^(1/3) cm. Option B (5π) would give volume 125π³, Option C (125π) would give volume (125π)³, and Option D (125) would give volume 1,953,125 - all incorrect.

Volume of region between cylinders = π(R² - r²)h = π(10² - 5²)(20) = π(100 - 25)(20) = π(75)(20) = 1500π. This uses the formula for volume of a cylinder (V=πr²h) and subtracts the inner cylinder volume from the outer.

Volume of sphere = (4/3)πr³. This is derived from integration. Option C correctly states this formula.

Surface area of a cube is 6y^2. Given 6y^2 = 294, then y^2 = 49. Taking the square root gives y = 7.00 cm.

When the radius increases by 6%, the new radius becomes 1.06r. Area is proportional to the square of the radius (A = πr²), so the new area becomes π(1.06r)² = 1.1236πr². This represents a 12.36% increase over the original area. The percentage increase is calculated as (1.06² - 1) × 100 = 12.36%. This demonstrates how small percentage changes in radius compound when applied to area.

Using the triangle area formula Area = (1/2) × base × height, we can solve for base: base = (2 × Area) ÷ height. With Area = 36 cm² and height = 4 cm, base = (2 × 36) ÷ 4 = 72 ÷ 4 = 18 cm, not 16 cm as stated.

Let the side be s. The new area is (s+3)^2 and the old area is s^2. (s+3)^2 - s^2 = 33. Expanding gives s^2 + 6s + 9 - s^2 = 33, so 6s = 24, and s = 4. Distractors like 6 or 2 result in different area increases.

If diameter increases by 100%, it doubles (2x). Since area is proportional to the square of the diameter (or radius), the new area is 2 squared, or 4 times the original. A 4x increase in area represents a 300% increase (400% - 100% = 300%).

The largest square inside a circle has its diagonal equal to the circle's diameter. For radius 10 cm, diameter = 20 cm. If diagonal = 20, then side = diagonal/√2 = 20/√2 = 10√2. Area = side² = (10√2)² = 200 sq cm. The claimed answer B is correct. Option C (100) would be if side = 10, but the diagonal constraint makes the square larger.

Statement A: lh = 4, bh = 6, lb = 9. Volume = lbh. From these: (lh)(bh)(lb) = l²b²h² = 4×6×9 = 216, so lbh = 6. Statement A alone is sufficient. Statement B: l + b + h = 10 and 2(lb + bh + lh) = 38. This gives lb + bh + lh = 19. With l+b+h=10 and sum of pairwise products=19, we can find (l+b+h)² = l²+b²+h²+2(lb+bh+lh), giving sum of squares = 100-38 = 62. But we cannot uniquely determine l, b, h from sum and sum of squares alone. Statement A alone suffices.

The largest circle that fits inside a square is the inscribed circle, whose diameter equals the side of the square. With side 14m, radius = 7m. Area = πr² = π × 49 ≈ 154 sq m.

Area is proportional to the square of the radius. If the radius becomes 0.9r, the area becomes (0.9)^2 = 0.81 of the original area. The diminution is 1 - 0.81 = 0.19, or 19%.

In mathematics, any finite number multiplied by zero results in zero. Regardless of the complexity of the areas calculated in the first two parts of the expression, the final multiplication by zero makes the product 0.

This is a playful math puzzle combining formula knowledge with wordplay. The volume of a cylinder is pi × r² × h, which becomes pi × z² × a = pizza. This type of riddle tests both formula recall and creative thinking.

Area depends on the square of the radius (A = πr²). When radius becomes 0.9r (10% reduction), new area = π(0.9r)² = 0.81πr². The reduction is 1 - 0.81 = 0.19 = 19%. Option A (10%) would be true for circumference, not area. Option C (20%) ignores the squared relationship.