How Wide Should the Interval Be

An interviewer asks you to estimate an unknown quantity and to give an interval around it. The scoring is stated up front: you collect $\$10$ if the true value lands inside your interval, and you are charged $\$1$ for every standard deviation of width you use. Formally, if your belief about the quantity is normal with mean $\mu$ and standard deviation $\sigma$, and you quote $\mu \pm k\sigma$, your expected score is

$$10 \left( 2\Phi(k) - 1 \right) - 2k.$$

Which $k$ maximises the score, what coverage does it correspond to, and what does that tell you about how confident an interval a trader should volunteer?