89 cm wore. Cut and bent into square and circle. Circumference of circle when area is minimum.

Question

You have a wire 89 cm long. You wish to cut it into two pieces. One poece will be bent into the shape of a square. The other piece woll 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

Hey Ryan.

 

You don't know much about the shapes besides the fact that they're a square and a circle, and that the perimeters add up to 89cm.

 

Let's let

 

s = side length of the square

r = radius of the circle

 

You know the perimeter of a square with side-length s will be 4s, and the circumference of a circle with radius r will be 2*pi*r

 

Since your original wire was 89cm long, that means:

 

4s + 2*pi*r = 89

 

You also know the area of a square with side length s is s^2, and the area of a circle with radius r is pi*r^2.

 

Since the problem stated that A is the total area for both the square and the circle, that means:

 

A = pi*r^2 + s^2

 

So now you have two equations.

 

4s + 2*pi*r = 89

 

A = pi*r^2 + s^2

 

I don't like dealing with multiple variables, so I'm going to use the perimeter equation to solve for s in terms of r.

 

4s = 89 - 2*pi*r

 

s = 89/4 - 1/2*pi*r

 

Plug this result back into your area equation.

 

A = pi*r^2 + (89/4 - 1/2*pi*r)^2

 

A = pi*r^2 + 7921/16 - 89/4*pi*r + 1/4*pi*r^2

 

A = 5/4*pi*r^2 - 89/4*pi*r + 7921/16

 

This area equation is a parabola. The minimum will occur when r = -b/2a

 

b = -89/4 * pi

a = 5/4 * pi

 

So

 

r = (89/4 * pi) / (2*5/4*pi)

 

r = (89/4) / (10/4)

 

r = 89/10

 

This will be the radius of the circle when the area is minimum. However, your question asks for the circumference of the circle when the area is minimum.

 

C = 2*pi*r

 

C = 2*pi*89/10

 

C = 89pi/5

 

Hope this helps.