Question

a piece of wire 12 meters long is cut into two pieces.one piece is bent into the shape of an equilateral triangle and the other into the shape of a circle.how should the wire be cut so that the combined area of the two figures is as small as possible?

Answer 1

if the triangle has side s, it has
area = √3/4 s^2
perimeter = 3s

if the circle has radius r, it has
area = πr^2
perimeter = 2πr

so,
3s+2πr = 12
s = 4 - 2π/3 r

and we want to minimize the area

a = √3/4 s^2 + πr^2
= √3/4 (4 - 2π/3 r)^2 + πr^2
= (π^2/3√3 + π^2)r^2 - 4π/√3 r + 4√3

That's just a parabola, with vertex at

r = (√3-1)/π = 0.233
so, s = 4 - (2π/3)(√3-1)/π = 3.512

3s + 2πr = 12.0