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?
Answers
Steve
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
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