Optimization triangle and circle

Question

A 2m wire is cut into two pieces. One is bent to make a triangle and another is bent to make a circle. Find where the wire should be cut for each figure to minimize the the total area of both figures added together and then maximize as well. I already know the radius as 2-x/2pi and the sides of the triangle as x/3. I need guidance from here. Help is appreciated!

Answer 1

Let x is length of circle and (2 - x) is lperimeter of triangle. Then aria of circle is π(x/2π)² = x²/(4π) and

area of triangle is [(2-x)/3]²√3/4 = (2 - x)²√3/36.

Total area A(x) = x²/(4π) + (2 - x)²√3/36; A(x)' = x/(2π) + (x - 2)√3/18 = 0; 9x + xπ√3 = 2π√3;

x = 2π√3/(9 + π√3) 0.7536 m

And this is min because A'(x) = 1/(2π)+ √3/18 > 0