Covering maximal number of sets using fixed number of elements
Your problem is essentially Minimum $k$-Union. In this problem (switching from $k$ to $\ell$), you want to find $\ell$ sets out of $A_1,\ldots,A_t$ which together cover the least number of elements. Denoting by $f(\ell)$ the solution of this problem and by $g(j)$ the solution of your problem, we have
$$
f(\ell) \leq j \Longleftrightarrow \ell \leq g(j).
$$
In other words, $g(j)$ is the maximal $\ell$ such that $f(\ell) \leq j$.

Unfortunately, Minimum $k$-Union is NP-hard. You can find some links in this question on StackOverflow.