Optimization

[Mika]

A piece of string whose length is 32 cm is cut into two pieces. One piece is used to form an equilateral triangle and the other to form a circle. What should be the perimeter of the equilateral triangle and the circumference of the circle so that the sum of the areas is a minimum?
Find the minimum sum of the areas. Express your answer in terms of pi.

 

[lex]

Start e.g. by letting r cm be the length of the radius of the circle. Work out the length of the side of the triangle.
Then finally write down an expression (in terms of r) for the exact total area = area of the triangle + area of the circle.
Then optimise this.

 

[Steven G]

Lex described a great method.
I would cut the string into 2 pieces, one being c cm and the other piece being 32-c cm.
Let c = the circumference of the circle and 32-c = perimeter of the equilateral triangle.

Now find the area of the circle and triangle in terms of c.