jmc

A cone with radius $r$ and height $h$ has volume $\frac{1}{3}\pi r^{2}h$; here, our cone has volume $\frac{1}{3}\pi (1^2)(4)=\frac{4}{3}\pi$. A sphere with radius $r$ has volume $\frac{4}{3}\pi r^3$, so we set up the equation $$\frac{4}{3}\pi r^3=\frac{4}{3}\pi .$$ Solving for $r$ yields $r^3=1$, so $r=1$. The sphere's radius is $\boxed{1}$ inch.