A piece of wire is to be cut into two pieces (one bent into
the shape of a regular p-gon and the other a regular q-gon).
If
1) p = 2*q,
2) 2*perimeter(q-gon) = 3*perimeter(p-gon), and
3) the sum of the two areas is minimized;
then what is the value of q?
I think I agree with Charlie here. From (2) we get that the p-gon
perimeter is 60% of the wire, and the q-gon perimeter is the other 40%.
Whatever p and q, we can always find a radius such that we get a p-gon
or q-gon with the given perimeter. So, all we need is to minimize the
sum of their areas. From (1), the q-gon is at least a triangle, and the
p-gon is at least an hexagon. If x<y, an x-gon is smaller than a
y-gon. So, if we take both minimums (q=3, p=6) we satisfy (3).
Agree
