Question

A wire of length 18 m is used to form a circle and a square. Find the radius of the circle and the side length of the square that produce a minimum total area?

Answer 1

• `{(r approx 1.26m ),(s approx 2.52m ):}`

Explanation:

If the wire of length is split into 2 lengths:

• `l_1, l_2 qquad " with "l_1 + l_2 = 18`

... to make circle and square respectively.

Circle

Circle will have circumference `C`:

• `C = 2 pi r = l_1 implies r = l_1/(2 pi )`

• `implies A_1 = pi r^2 = l_1^2/(4 pi )`

Square

Square will have perimeter `p` with side `s`:

• `p = 4 s = l_2 implies s = l_2/4`

• `implies A_2 = s^2 = l_2^2/16`

Total Area

`A(bbl) = A_1(l_1) + A_2(l_2) = l_1^2/(4 pi ) + l_2^2/16`

`= l_1^2/(4 pi ) + (18 - l_1)^2/16`

This is now in single variable. Differentiate wrt `l_1` to optimise:

`A' = l_1/(2 pi) - (18 - l_1) /8 = 0`

`implies {(l_1 = (18 pi)/(4 + pi) ),(l_2 = 72/(4 + pi) ):} qquad {(r approx 1.26m ),(s approx 2.52m ):}`

Using approximate values for `l_1,l_2`:

`A approx 11.3 \ m^2`

The 2nd derivative is:

`A'' = 1/(2 pi)+ 1/8 > 0`

So this is a minimum .