Optimal threshold in a song-stopping bet

A song lasts $234$ seconds. You name a number $x$ with $0<x<1$; then the song is halted at a time chosen uniformly over its length. If it halts no later than $234x$ seconds in, you lose $2x$ units; if it halts after that, you win $x$ units. Which $x$ gives you the greatest expected profit?

$x=\tfrac16$. You lose $2x$ with probability $x$ and win $x$ with probability $1-x$, so your expected profit is $x-3x^2$, a downward parabola. Its vertex is where $1-6x=0$, i.e. $x=\tfrac16$, giving profit $\tfrac1{12}$.