Question

A piece of wire 24cm long is cut into two pieces. One piece is bent to form a circle, the other forms a square. What is the length of each piece if the total area of the two shapes is a minimum?

Answer 1

Call L the length used to make the circle - it'll be the circumference of the circle and circumference, L = 2*pi*r,

so r = L/(2pi),

area of the circle is pi*r^2, so: pi*(L/(2pi))^2


Since the total length is 24cm, the length used for the whole square is 24-L, and so the length of just one side of the square is (24-L)/4

Area of a square is the side length squared... ((24-L)/4)^2

Total area is the sum of the circle and square \[\Large A = \pi \left( \frac{ L }{ 2 \pi } \right)^2 + \left( \frac{ 24-L }{ 4 } \right)^2\]

Answer 2

Now you have to start the calculus part of the question - differentiate it, set it equal to zero to find the minimum, and solve for L.

Since it's a quadratic you can also find the minimum value without calculus...

Answer 3

Thanks for the reply, helped me out. :)