If Paul is using a piece of fencing 80 meters long to build a rectangular enclosure for his dog, what is the greatest possible area that can be enclosed?

Long way:

Say the sides are x and y long. The perimeter of the fence must be 80, so 2x + 2y = 80. The area of the fence will be xy.

We can use the first equation to get the second equation in terms of only one variable!

2x + 2y = 80
y = 40 – x

Now you can say that the area of the rectangle will be x(40 – x). You can graph that if you want, or you can just recognize that it will be a downward facing parabola with roots at 0 and 40. That means it’ll have its maximum when x = 20. That means the greatest possible area will be when x = 20, when the area will equal 400 square meters.


Just remember that in a case like this, the biggest rectangular area you can enclose is with a square. If you know the perimeter needs to be 80, then make a 20×20 square.

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