jmc

We have a geometric sequence with first term 1000 and common ratio $1/2$. Any term in this sequence can be represented as $1000\cdot {\left(\frac{1}{2}\right)}^k$, where $k$ is the number of bounces (for example, when $k=1$, $1000\cdot {\left(\frac{1}{2}\right)}^k=500$, or the height of the $k=1^{\text{st}}$ bounce). We need to find the smallest $k$ such that $1000\cdot {\left(\frac{1}{2}\right)}^k<1$. Through trial and error, we find that $k=10$, so it takes $\boxed{10}$ bounces for the maximum height to be less than 1 foot.