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

Options

A. `2x=(pi+4)r`
B. `(4-pi)x=pir`
C. `x=2r`
D. `2x=r`

Answer

C

Explanation

Arroding to given information, we have Perimeter of square+ Perimeter of circle =2 uinits
`rArr 4x+2pi 3=2`
`rArr r=(1-2x)/(pi)" "...(i)`
Now, let A be the sum of the areas of the squre and the circle. Then,
`A=x^(2)+pi r^(2)`
`=x^(2)+pi((1-2x)^2)/(pi)`
`rArr A(x)=x^(2)+((1-2x)^(2))/(pi)`
Now, four minimum value of `A(x), (dA)/(dx)=0`
`rArr 2x+(2(1-2x))/(pi)*(-2)=0 rArr x=(2-4x)/(pi)`
`rArr pi x+4x=2 rArr x=(2)/(pi+4)" "....(ii)`
Now, from Eq. (i), we get
`r=(1-2*(2)/(pi+4))/(pi)=(pi+4-4)/(pi(pi+4)) = (1)/(pi+ 4)" "....(iii)`
From Eqs. (ii) and (iii) , we get x=2r