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?

The idea is to minimize the sum of the two areas by choosing q, which must be at least 3, making p at least 6.

The total perimeter is fixed, equal to the total length of the piece of wire.  The q-gon gets 3/5 of the length of the wire, while the p-gon gets 2/5.

As the number of sides is increased, with the same given perimeter, the area enclosed increases, so we want to make the number of sides as small as possible: that makes the value of q 3.

Posted by Charlie on 2005-05-20 19:15:03