Question

A wire of length $28m$ is to be cut into two pieces one of the pieces is to be made into a square and other into a circle. What should be the length of two pieces so that the combined area of the square and circle is min?

Answer 1

Let the sides be $x,28-x$

Area of square = $\frac{(28-x{)}^2}{16}$

area of circle = $\pi \times r^2=\frac{x^2}{4\pi }$

combined area,A = $\frac{x^2}{4\pi }+\frac{(28-x{)}^2}{16}$

$dA=\frac{x}{2\pi }-\frac{(28-x)}{8}$

$0=4x-\pi (28-x)$

$x=\frac{28\pi }{4+\pi }$

Other part, $28-x=\frac{112}{4+\pi }$