Question 976989
Take any wire of length 16 inches. Cut the wire into two pieces such that the sum of the areas of the square and circle formed by the pieces of wire is the minimum.
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s = side length of the square.
Circumference of the circle = 16 - 4s = 2pi*r
r = (16-4s)/2pi = (8-2s)/pi
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Area of circle = pi*(8 - 2s)^2
Area of square = s^2
Total = s^2 + pi*(8 - 2s)^2
dA/ds = 2s + pi*2*(8-2s)*(-2) = 2s - 4pi*(8-2s) = 0
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2s - 4pi*(8-2s) = 0
s = 16pi - 4pi*s
s*(4pi + 1) = 16pi
s = 16pi/(4pi + 1)
s =~ 3.705 inches
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dA/ds = 2s - 4pi*(8-2s)
2nd derivative = 2 + 8pi which is positive --> a minimum
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You can do it using dA/dC in a similar manner.
I prefer making the side of the square the variable.