Question

She wants to create a rectangular area of 1,600 square feet for her pet and can afford to purchase 160 feet of fence. In two or more complete sentences, explain the algebraic model, calculations and reasoning necessary to determine the dimensions of the rectangular area.

Answer 1

Explanation:

"Rectangular" includes "square," and in fact treating the rectangle as a square is the fastest way to "solve" this problem. The perimeter of the rectangular area is 160 ft; dividing that by 4 yields 40 ft. "40 ft square" describes the rectangle maximum area: (40 ft)^2 = 1600 ft^2.

Alternatively, let W and L represent the width and length of the rectangle. Then:

P = perimeter = 2W + 2L = 160 ft, or W + L = 80 ft, or W = 80 ft - L.

A = area = L*W = L(80 ft - L) = 1600 ft^2. Rewriting this as a proper quadratic:

80L - L^2 - 1600 = 0, or L^2 - 80L + 1600. Note that this last result factors into (L - 40)^2 = 0, so L = 40 ft. Then W = 80 ft - 40 ft = 40 ft.

This confirms that the max area is 1600 ft^2 = (40 ft)^2.