The area of wastage when cutting a circular sheet from a rectangular sheet is 546 m² and the perimeter of the rectangular sheet is 128
m. If the length of the rectangular sheet is increased by 20%, find the new area of wastage. (The breadth of the rectangle is equal to the diameter of the circle) Note: The breadth of the rectangular sheet is a positive integer (no decimals).

ATQ, Let length = L and breadth = B. Perimeter = 2(L + B) = 128 ⇒ L + B = 64 ⇒ L = 64 − B Diameter of circle = B Radius = B/2 Wastage = Area of rectangle − Area of circle 546 = LB − (22/7) × (B/2)² 546 = (64 − B)B − (22/7) × B²/4 ⇒ 546 = 64B − B² − (11/14)B² Multiply both sides by 14: 546×14 = 14×64B − 14B² − 11B² 7644 = 896B − 25B² Rearrange: 25B² − 896B + 7644 = 0 Check B = 14: 25×196 − 896×14 + 7644 = 4900 − 12544 + 7644 = 0 So, B = 14 m Then, L = 64 − 14 = 50 m New length = 120% of 50 = 1.2 × 50 = 60 m New rectangle area = 60 × 14 = 840 m² Area of circle = (22/7) × (14/2)² = (22/7) × 7² = (22/7) × 49 = 154 m² New area of wastage = 840 − 154 = 686 m²