Let
s = side of square
r = radius of circle
We know that
4s + 2πr = 65
Now, the combined area
A = s^2 + πr^2 = ((65-2πr)/4)^2 + πr^2 = (π^2 + 4π)r^2 + 65π/4 r + 4225/16
This is just a parabola, with its vertex at r = -b/2a = 65/(2π+8)
so the value of A = 4225/(4π+16)
You have a wire that is 65 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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