What Is the Shape of a Mylar Balloon?

Mylar balloons of different shapes and sizes have become popular as gifts or in bouquets. The most common balloons are comprised of two circular sheets of mylar, fused together at the circumference. A small opening on the circumference allows the balloon to be inflated with either air or helium. These balloons are not spherical, which is at first surprising, for it is well known that the sphere gives the maximal volume for a given surface area. Thus, these balloons suggest the following mathematical problem: Given a circular mylar balloon of deflated radius r, what will be the shape of the balloon when it is fully inflated? In particular, we could ask