Question

Show that of all the rectangles with a given perimeter,the square has the largest area.

Answer 1

If one side of a square is a, then,

perimeter = 4a,

which is fixed, and,

area = a².

Let one side of the square change by some distance, say x. So if it is increased by x, then the square will be changed to rectangle and this side becomes the length, and breadth is formed by reducing its adjacent sides by the same distance x, since perimeter is constant.

I.e., if one side a is changed to (a + x), then this will be the length, and breadth will be (a - x). So we get a rectangle.

Here perimeter does not change.

2((a + x) + (a - x)) = 2(2a) = 4a

But area changes,

(a + x)(a - x) = a² - x².

Here we can see that, as the change in side of the square, i.e., x, increases, the area becomes very smaller.

So the area of the rectangle formed by changing the side of the square by 2 units is much lesser than that when the side is changed by 1 unit.

Also, from this, we get that the maximum area of the rectangle is got when there's no change in the side of the square, i.e., when x = 0. This implies maximum area is given by the square for a given perimeter.

Hence Proved!