Question
Mordecai has a large target of radius $10$ and a laser pointer. He is clumsy, so he points the laser at a uniformly random point on the target. Let $D$ represent the random distance from the laser to the center of the target. He offers you to play the following game: Mordecai points the laser at the target one time. Before doing so, you fix a value $0 \leq r \leq 10$. If your laser is within distance $r$ away from the center, you must pay Mordecai $\$(10-r)$. Otherwise, Mordecai pays you $\$r$. Find the value of $r$ that maximizes your expected winnings from this game.