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.
• B.
• C.
• D.

Answer 1

The correct option is B

Length of wire = 2

$Given4x+2\pi r=2\Rightarrow 2x+\pi r=1......\left(i\right)A=x^2+\pi r^2={\left(\frac{1-\pi }{2}\right)}^2+\pi r^2\Rightarrow \frac{dA}{dr}=2\left(\frac{1-\pi r}{2}\right)(-\frac{\pi }{2})+2\pi r\text{For max and min}\Rightarrow \frac{dA}{dr}=0\pi (1-\pi r)=4\pi r1=4r+\pi r........\left(ii\right)\text{From (i) and (ii)}2x+\pi r=4r+\pi rx=2r$