HMMT 2001 K9

On the planet Lemniscate, the people use the elliptic table of elements, a far more advanced version of our periodic table. They're not very good at calculus, though, so they've asked for your help. They know that Kr is somewhat radioactive and deteriorates into Pl , a very unstable element that deteriorates to form the stable element As. They started with a block of Kr of size 10 and nothing else. (Their units don't translate into English, sorry.) and nothing else. At time $t$, they let $x(t)$ be the amount of $\mathrm{Kr}, y(t)$ the amount of Pl , and $z(t)$ the amount of As. They know that $x^{\prime}(t)=-x$, and that, in the absence of $\operatorname{Kr}, y^{\prime}(t)=-2 y$. Your job is to find at what time $t$ the quantity of Pl will be largest. You should assume that the entire amount of Kr that deteriorates has turned into Pl .