Question

Patricia purchased x meters of fencing. She originally intended to use all of the fencing to enclose a square region, but later decided to use all of the fencing to enclose a rectangular region with length y meters greater than its width. In square meters, what is the positive difference between the area of the square region and the area of the rectangular region?

Answer 1

(y^2)/4 square meters

Explanation:

For a perimeter length of x, the side of a square will be x/4 and its area will be (x/4)^2.

If one side of the square is shortened by y/2 and the adjacent side is lengthened by y/2, then the difference in side lengths will be y. The area of the resulting rectangle will be ...

(x/4 -y/2)(x/4 +y/2) = (x/4)^2 -(y/2)^2

That is, the difference in area between the square and the rectangle is ...

(x/4)^2 - ((x/4)^2 -(y/2)^2) = (y/2)^2 = y^2/4

The positive difference between the area of the square region and the area of the rectangular region is y^2/4 square meters.