Circumference of circle + Perimeter of Square = 60



Solve for r



Divide through by 2



Multiply both sides by 



Let y = Area of circle + Area of square

    

We need to substitute for 




Substituting:








[If you were taking calculus you would take the derivative here and set it
equal to 0, but I am assuming that you are taking college algebra, so you must
use the vertex formula.]







or in descending powers:



The coefficient of  is positive so this represents
a parabola that opens upward, so its vertex will be at a minimum

To find the x-cordinate of the vertex, we use the vertex formula

x-coordinate of vertex = 

x-coordinate of vertex = 


So for the minimum area, the side of a square will be  cm.
That is approximately 8.401487303 cm.

We will need to cut the wire at 4 times the side of the square.

So we must cut the wire at  or  
or about 33.60594921 cm from one end and, subtracting from 60,
26.39405079 cm from the other end.

Now for the maximum area. The problem is only defined for 
When x=0, the square shrinks to 0 and the whole 60cm wire is made into a
circle.  When x=15, making the perimeter of the square 60 cm, the circle
shrinks to 0 and the whole 60cm wire is made into a square.  Since the parabola
opens upward, the maximum value is at one endpoint of the interval, either when
x=0 or when x=15.  It is well known that if a piece of wire is bent into a
circle or a square, the circle will have more area, so we could just assume the
maximum area would be when we "cut" the wire 0, or no, centimeters from the
end, and bend the whole wire into a circle. That is we don't cut the wire at
all.

Edwin