AHSME 1996 Problem 27

Consider two solid spherical balls, one centered at $\left(0, 0,\frac{21}{2}\right)$ with radius $6$, and the other centered at $(0, 0, 1)$ with radius $\frac{9}{2}$. How many points with only integer coordinates (lattice points) are there in the intersection of the balls? $\text{(A)}\ 7 \text{(B)}\ 9 \text{(C)}\ 11 \text{(D)}\ 13 \text{(E)}\ 15$