A wire of length 2 units is cut into two parts which are bent respectively to form a squar…

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:

Accepted Answer

$$4x + 2\pi r = 2$$ $$\,\,\,$$ $$ \Rightarrow 2x + \pi r = 1$$

$$S = {x^2} + \pi {r^2}$$

$$S = {\left( {{{1 - \pi r} \over 2}} \right)^2} + \pi {r^2}$$

$${{dS} \over {dr}} = 2\left( {{{1 - \pi r} \over 2}} \right)\left( {{{ - \pi } \over 2}} \right) + 2\pi r$$

$$ \Rightarrow {{ - \pi } \over 2} + {{{\pi ^2}r} \over 2} + 2\pi r = 0$$

$$ \Rightarrow r = {1 \over {\pi  + 4}}$$

$$ \Rightarrow x = {2 \over {\pi  + 4}}\,$$

$$ \Rightarrow x = 2r$$

Solutions · 1