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 Question from Johann, a student: A gardener has 140 ft of fencing to fence in a rectangular shaped vegetable garden. To plant all his vegetables, the gardener needs mora than 825 sq ft of space. What are the possibilities values for the length of the rectangle that would meet the space requirements? You may assume that the variable x represents the length of the rectangle.

Hello Johann. I will solve a similar problem and show you how you can do this with your own question. Say there is 130 meters of fencing and you need more than 1000 square meters of space in a rectangular shape.

The area of a rectangle is the length times the width: LW = A. So if the area is MORE 1000, then

LW > 1000.

You also know that the fence goes around the perimeter of the rectangle, so

L + L + W + W = 130.
2L + 2W = 130
L + W = 65
L = 65 - W

Now, we can use this last line to substitute L with (65 - W) in the earlier inequality like this:

LW > 1000
(65 - W)W > 1000

multiply by -1 on both sides (this reverses the inequality)

(W - 65)W < -1000
W2 - 65W + 1000 < 0

This is a quadratic equation. You can solve it using the quadratic formula, but it is easier in this case to solve it by factoring:

(W - 45)(W - 20) < 0

So W must be between 20 and 45. Since L = 65 - W, we have L between 20 and 45 as well.

Hope this example helps you,
Stephen La Rocque.

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.