Find the dimensions of the rectangular corral producing the greatest enclosed area given 340 feet of fencing.

Answer 1

Hi Hannah!

With a set amount of fencing, the biggest rectangular area that you can enclose will always be a square! (There's a way to prove this with logic, but at the end I'll show you a way to prove it through guess and check)

If the rectangle is a square, then all four sides are the same length, because that's part of what it means to be a square, and if the perimeter of the square must be 340 feet, then each side of the square must be

340/4 = 85 feet.

Then, if the length and width of the square are 85 feet, the area of the square is A = L×W = 7225 ft ² .

Now, how do we know that's the largest area it could be? Why not try making the rectangle just barely not square, and see the new result? For instance, what if we made the rectangle have dimensions of 84 ft × 86 ft?

The perimeter would still add to equal 340 ft (86 + 86 + 84 + 84 = 340), but when we go to calculate the area, A = L × W = (84) × (86) = 7224 ft² which is one less than the answer we got for a perfect square of 7225 ft²! And sure enough, if you carry on in much the same way you will find the area always decreases more and more.

Hope this was helpful!