Area of circles and squares

Question

A wire that is 5 feet long is to be used to form a square and a circle. How much of the wire should be used for the square and how much for the circle to enclose the maximum area?

Answer 1

Let's assume the length of each side of the square is x feet

the circle to enclose the maximum area should inscribe the square and thus should have a diameter = diagonal of the square = x√2 feet

 

Thus the perimeter of the square would be = 4x feet

and the periphery of the circle would be = πx√2 feet

 

Given that,

4x+πx√2≤5

=> x(4+π√2)≤5 .... adjusting operators

=>x≤5/(4+π√2) ... dividing both sides by (4+π√2)

 

The maximum length of the wire required to make the square= 4x feet = 4*5/(4+π√2) feet = 20/(4+π√2) feet≈2.37 feet

The maximum length of the wire required to make the square=πx√2 feet=π*√2*5/(4+π√2) feet =≈2.63 feet

 

Around 2.37 feet of the wire should be used for the square and nearly 2.63 feet of the wire should be used for the circle to enclose the maximum area.