When to stop rolling a ten-sided die that can bust

You repeatedly roll a fair ten-sided die with faces $1$ to $10$, deciding before each roll whether to continue. Any result from $1$ to $9$ is added to your winnings; a $10$ wipes out everything you have accumulated. Under optimal play you should stop as soon as your total reaches at least $\text{\char36}k$. What is $k$?

$45$. With total $k$, one more roll keeps $k$ plus an average gain of $5$ with probability $9/10$ and loses all otherwise. Rolling is worth it while $0.9(k+5)>k$, so the break-even point is $k=45$.