Question

a=400mmd=400mm wastage=?​

Answer 1

Answer:

s worthwhile to know this theorem: Among all rectangles with a given perimeter, the square has the largest area.

You can apply that to this question, and conclude that the square of side 100 mm has the largest area of all rectangles with perimeter 400 mm.

This theorem is typically an exercise in differential calculus when studying maxima and minima. It shows how easy it is to solve problems like this using calculus.

But as it’s a quadratic problem, calculus isn’t necessary to answer the question. The theorem was known in antiquity, and the ancient Greek geometers could prove it. You can show it directly using geometry or symbolic algebra.

Here’s a proof using symbolic algebra. Let PP denote the perimeter of a rectangle. Since the sum of two adjacent sides is P/2,P/2, call one side P/4+xP/4+x and the adjacent side be P/4−x.P/4−x. Then the area of the rectangle is their product, and that product is equal to P2/16−x2.P2/16−x2. Now the square with that perimeter has all sides equal to P/4,P/4, and the area of that square is P2/16.P2/16. Since the area of that square is greater than or equal to P2/16−x2,P2/16−x2, therefore the square has the greatest area. Q.E.D.

This is the oldest isoperimetric problem. An isoperimetric problem is one in which you want to find the figure among a given class of figures with the same perimeter having the greatest area. Another isoperimetric problem is to find the largest triangle with a given perimeter. That’s a harder problem, and the answer, as you might expect, is an equilateral triangle.