Q. A wire of length 36 units is cut into two parts which are bent respectively to form a square of side x units and a circle of radius of r units. If the sum of areas of the square and the circle so formed is minimum, then find the circumference of the circle.
Answer:
2π x 18/(4 + π) = 36π/(4 + π)
Solution:
Sum of perimeter of square and circle = 36
4x + 2πr = 36⇒2x + πr = 18
x = (18 – πr)/2 = 9 – πr/2 …..(i)
Sum of areas: S = x2 + πr2 ….(ii)
S = πr2 + (9 – πr/2) 2 , diff. w.r.to r
dS/dr = 2πr + 2(9 – πr/2)( – π/2)
dS/dr = 0
⇒r – 9/2 + πr/4
⇒r(π + 4)/4 = 9/2
⇒r = 18/(4 + π)
Circumference of circle = 2π x 18/(4 + π) = 36π/(4 + π)