A wire of length L can be used to make a circle and a square. How much of the wire should be used for each of the figures to maximise area?
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The total length of the wire is L. Let us use it to make a square of length x and a circle of radius r. The combined area of the shapes is x^2 + pi*r^2. The circumference of the square is 4x and that of the circle is 2*pi*r
4x + 2*pi*r = L
We have to maximize A = x^2 + pi*r^2
Differentiate L and A with respect to r
dA/ dr = 2*x(dx/dr) + 2*pi*r
dL / dr = 4(dx/ dr) + 2*pi
As L is a constant dL/dr = 0
=> 4(dx/ dr) + 2*pi = 0
=> dx / dr = -2*pi/4 = -pi/2
substitute in dA/dr
=> dA/ dr = 2*x(-pi/2) + 2*pi*r
=> dA/dr = -pi*x + 2*pi*r
Take the second...
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