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?

Question

Mathematics

Elena Carey

2 July, 09:44

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?

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Answers (1)

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Evan Lam · Answer 1

(y^2) / 4 square meters

Step-by-step 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.