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?

Explanation

Let r be the radius of the circle and x be the side of square.
`:. ` Circumference of circle + perimeter of square = 28 cm
` rArr 2 pi r + 4x =28`
`rArr x = (14 - pi r)/2 ` ….(1)
and let area of circle + area of square = A
`:. A = pi r^(2) + x^(2)`
` = pi r^(2) + ((14 - pi r)/2)^(2)` [From equation (1)]
` = pi r^(2)+ 1/4 (pi^(2)r^(2)- 28 pi r+ 196)`
` rArr (dA)/(dr) = 2 pir+1/4(2pi^(2)r-28pi)`
and ` (d^(2)A)/(dr^(2) = 2 pi + 1/4 (2pi^(2))`
For maxima/minima ` (dA)/(dr) = 0`
` rArr pir+1/4 (2pi^(2)r-28pi)=0`
` rArr pir+pi^(2)r-14pi =0`
` rArr r(4+pi)=14`
` rArr r=14/(pi+4)`
at ` r=14/(pi+4)`
` (d^(2)A)/(pi r^(2)) gt 0`
` rArr ` A is minimum.
Now ` 2pi r = 2pi((14)/(pi+4)) = (28pi)/(pi+4)`
and ` 4x = 28 - 2pi r `
` = 28 - (28pi)/(pi+4) = (112)/(pi+4)`
Therefore, the length of two pieces of wire ` = (28pi)/(pi+4) cm and (112)/(pi+4)` cm.