The Art of Mathematics

I found this interesting question in one of the articles I was reading on Calculus,

“A rectangle is to be formed by using a piece of wire that is 36 inches long. What will its dimensions be if it encloses the greatest area?”

Here, we have a problem that states, the perimeter of a rectangle is 36 inches, and we have to deduce the dimensions of that rectangle (length and breadth) which will have the greatest area possible.

I have been trying to calculate that.

To cut the long story short, I have made the following table, given the fact that everybody knows that the formula.

Perimeter of a rectangle = 2 (l + b) units

Area of a rectangle = l X b sq. units

With the above conditions, the possible dimensions of the length and breadth should have a sum of l + b = 18

If, 2 (l + b) = 36,

Then, l + b = 18

So the possible dimensions are, 17 + 1, 16 + 2, and 15 + 3… so on

Table: Perimeter v/s Area of the rectangle

P=2(l+b)	36	36	36	36	36	36	36	36	36
l+b	18	18	18	18	18	18	18	18	18
A=lXb	17	32	45	56	65	72	77	80	81
l	17	16	15	14	13	12	11	10	9
b	1	2	3	4	5	6	7	8	9

I have even plotted a graph for that Perimeter against Area. 

With Perimeter being constant at 36 inches, the greatest area is achieved at 81 sq. inches, and the dimensions of length and breadth is 9 inches.

Here is the interesting part, it makes a square. Yes, the rectangle with the greatest area is a ‘square’.

I tried working out for other possible values of perimeter as well. For every problem, it comes that “area with the greatest rectangle is a square, for the given value of perimeter.”

So what?

If you  have constructing or living in a home, we could now deduce that best possible room with the maximum amount of space/ area will be a square with the same amount of mortar used (perimeter value).

But we don’t usually build square rooms, do we?

On the other hand, it has been a very common practice among us, as told by our forefathers, the best possible dimensions of a room will be a product of two consecutive odd numbers, say – 9 X 11, 11 X 13… etc… This has serious amount of scientific reasoning into it, for the product of two consecutive odd numbers is the ‘second best’ spacious dimension for the given amount of perimeter.

Interesting isn’t it?

So two conclusions that has kept me thinking were,

1. The possibility that there could be two rectangles with same perimeter but varying area
2. A square is the largest rectangle with maximum area for the given perimeter 

Oh the beauty of Maths!!