Question

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum, then:

• A. $x=2r$
• B. $2x=r$
• C. $2x=(\pi +4)r$
• D. $(4-\pi )x=\pi r$

Answer 1

The correct option is A $x=2r$

$4x+2\pi r=2\Rightarrow 2x+\pi r=1$
$S=x^2+\pi r^2$
$S={\left(\frac{1-\pi r}{2}\right)}^2+\pi r^2$
$\frac{dS}{dr}=2\left(\frac{1-\pi r}{2}\right)\left(\frac{-\pi }{2}\right)+2\pi r$
$\Rightarrow \frac{-\pi }{2}+\frac{{\pi }^{2}r}{2}+2\pi r=0\Rightarrow r=\frac{1}{\pi +4}$
$\Rightarrow x=\frac{2}{\pi +4}\Rightarrow x=2r$