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?

Question

Mathematics

Sidney Arnold

17 May, 10:09

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?

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Corinne Obrien · 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