A rectangle and a square each have a perimeter of 160 meters. The rectangle's area is 100 square meters less than the area of the square. What is the length of the rectangle?

Explanation

Since the perimeter of the square is 160 meters, each side measures 160 ÷ 4 = 40 meters. For the rectangle, the perimeter formula 2(l + b) = 160 gives l + b = 80. The square's area is 40 × 40 = 1600 square meters. The rectangle's area is 100 square meters less, so lb = 1600 - 100 = 1500. Using (l - b)^2 = (l + b)^2 - 4lb = 80^2 - 4 × 1500 = 6400 - 6000 = 400, so l - b = 20. Solving the system: l + b = 80 and l - b = 20, we get 2l = 100, hence l = 50 meters.