This question tests finding area given circumference of a circle. The circumference formula is C = 2πr, so with C = 18π, we have 2πr = 18π, giving r = 9 cm. The area formula is A = πr², so A = π(9)² = 81π square centimeters. This matches answer choice E. A common error would be using the radius value directly without squaring it, giving A = 9π (choice A), or confusing radius with diameter calculations.

This question tests percent change in area when dimensions change. For a square with side length s, the area is A₁ = s². When the side length increases by 50%, the new side length is 1.5s, and the new area is A₂ = (1.5s)² = 2.25s². The percent increase in area is [(2.25s² - s²)/s²] × 100% = [1.25s²/s²] × 100% = 125%. This matches answer choice D. A common error is assuming that a 50% increase in side length yields a 50% increase in area (choice A), failing to account for the quadratic relationship between linear dimensions and area.

This question tests the area formula for a circle. The area of a circle with radius r is A = πr², so with radius 7 cm, the area is A = π(7)² = 49π square centimeters. This formula represents the fundamental relationship between a circle's radius and its enclosed area. The area grows quadratically with the radius, not linearly. A common mistake is to use the circumference formula 2πr instead, which would give 14π, or to forget to square the radius and calculate 7π.

This question tests the volume formula for a right circular cylinder. The volume of a cylinder is V = πr²h, where r is the radius and h is the height. With radius 3 inches and height 10 inches, the volume is V = π(3)²(10) = π(9)(10) = 90π cubic inches. This formula represents the area of the circular base (πr²) multiplied by the height. A common error is to use the diameter instead of the radius, which would give π(6)²(10) = 360π, or to confuse the formula with surface area calculations.

This question tests the relationship between area and perimeter of a square. For a square with area A, each side has length s = √A, so with area 81, each side is √81 = 9 units. The perimeter of a square is P = 4s = 4(9) = 36 units. The key insight is that area equals side squared, so we must take the square root to find the side length before calculating perimeter. A common error is to confuse area and perimeter formulas, perhaps dividing 81 by 4 to get 20.25 or multiplying 81 by 4 to get 324.

This question tests how area scales when dimensions change by a percentage. If each side of the square is decreased by 25%, the new side length is 20 × (1 - 0.25) = 20 × 0.75 = 15 cm. The area of the new square is 15² = 225 square centimeters. Alternatively, since area scales as the square of the linear scaling factor, the new area is 400 × (0.75)² = 400 × 0.5625 = 225. A common error is to decrease the area by 25% directly, calculating 400 × 0.75 = 300, which incorrectly applies the percentage decrease to the area rather than to the linear dimensions.

This question tests the volume of a cone. The volume is (1/3)πr²h. Radius 3 cm, height 12 cm gives (1/3)π×9×12 = (1/3)π×108 = 36π cubic centimeters. This applies the formula directly. The result is justified by the cone's volume principle. A distractor like 108π might omit the 1/3 factor. Another like 432π could multiply extra.

This question tests the area of a square given its perimeter. The area of a square is side², and the side length is perimeter divided by 4. With a perimeter of 48 inches, the side is 48 / 4 = 12 inches, so the area is 12 × 12 = 144 square inches. This applies the relationship between perimeter and side directly. The result is justified as it follows from the square's properties. A distractor like 576 might result from squaring half the perimeter (24² = 576) due to a scaling misconception. Thus, the area is correctly 144.

This question tests the area of a circle given its circumference. The circumference is C = 2πr, so r = C / (2π); area is πr². Given C = 30π cm, r = 30π / (2π) = 15 cm, so area = π×15² = 225π square centimeters. This derives radius first then applies the area formula. The result is justified by the relationship between circumference and radius. A distractor like 900π might come from squaring 30 instead of 15. Another like 60π could be from using diameter incorrectly.

This question tests how area scales when dimensions change. The original rectangle has area = 9 × 4 = 36 square feet. When both length and width are doubled, the new dimensions are 18 ft × 8 ft, giving new area = 18 × 8 = 144 square feet. The area increases by a factor of 4 (2² = 4) when both dimensions double, confirming 36 × 4 = 144, which matches choice B. A common error would be thinking area only doubles (36 × 2 = 72, choice A) when dimensions double.