Probability - Rolling a Dice

A person rolls a pair of six-sided dice, which are equally likely to come up any number from 1 to 6. If the person rolls a seven he gets $0. If he rolls any other number, he can choose either to win that amount or play the game again from the start. This person believes that the best strategy is to keep rolling until he gets a "high" number, so he wins a lot of money. However, he doesn't know what "high" number to pick.
Let's say he will stop rollling if he rolls a K or more, otherwise he will roll again.

Problem 1: What value of K maximizes his e-winnings?

a) 4
b) 6
c) 8
d) 10

Problem 2: What is the standard deviation of your distribution of his winnings, if he plays the correct K above?

a) $2.81
b) $3.32
c) $4.35
d) $5.17