A wire of length L can be used to make a circle and a square. How much of the wire should be used for each of the figures to maximise area?

The total length of the wire is L. Let us use it to make a
square of length x and a circle of radius r. The combined area of the shapes is x^2 +
pi*r^2. The circumference of the square is 4x and that of the circle is
2*pi*r

4x + 2*pi*r = L

We have
to maximize A = x^2 + pi*r^2

Differentiate L and A with
respect to r

dA/ dr = 2*x(dx/dr) +
2*pi*r

dL / dr = 4(dx/ dr) +
2*pi

As L is a constant dL/dr =
0

=> 4(dx/ dr) + 2*pi =
0

=> dx / dr = -2*pi/4 =
-pi/2

substitute in
dA/dr

=> dA/ dr = 2*x(-pi/2) +
2*pi*r

=> dA/dr = -pi*x +
2*pi*r

Take the second derivative with respect to
r

=> d^2A / dr^2 = 2*pi –
pi*(dx/dr)

=> d^2A / dr^2 = 2*pi +
pi^2/2

The second derivative is always positive. The
function of A versus r is concave upwards.

In the interval
0 <= 2*pi*r <= L, the function A takes the maximum value at either of r =
0 or r = L or both.

At r = 0, x = L/4, we find A = L^2 /
16

At r = L/2*pi, x = 0, we have A = L^2/
4*pi

This gives the maximum value of A at r = L/
2*pi

Therefore we can conclude that for the
maximum area the wire should be used only to make the circle and no part of it used for
the square.