Question

You have a wire that is 29 cm long. you wish to cut it into two pieces. one piece will be bent into the shape of a square. the other piece will be bent into the shape of a circle. let a represent the total area of the square and the circle. what is the circumference of the circle when a is a minimum?

Answer 1

C = circumference of circle
p = perimeter of square
c + p = 29
p = 29 - c

side of square = (29 - c)/4
radius of circle = c/(2pi)

a = side^2 + (pi)r^2

a = ((29 - c)/4)^2 + pi * (c/(2pi))^2

a = (1/16)(841 - 58c + c^2) + (c^2/(4pi^2)) * pi

a = 52.5625 - 3.625c + c^2/16 + c^2/(4pi)

Since the 2nd degree term has a positive coefficient is positive, this has a graph that is a parabola opening up. It has a minimum value.

We take the derivative of the area expression and set equal to zero to find the minimum circumference.

da/dc = -3.625 + c/8 + (c/(2pi)) = 0

c = 12.76

Answer: 12.76 cm