Liz is considering a cellular phone service plan. Under this plan, she would specify a quantity of minutes, say x, per month that she would buy at 5 per minute. Hence, her upfront cost would be $0.05x. If her usage is less than this quantity x in a given month, she loses the minutes. If her usage in a month exceeds this quantity x, she would have to pay 40 for each extra minute (that is, each minute used beyond x). For example, if she contracts for x = 120 minutes per month and her actual usage is 40 minutes, her total bill is $120 0.05 = $6.00. However, if actual usage is 130 minutes, her total bill would be $120 0.05 + (130 120) 0.40 = $10.00. Liz estimates that her monthly needs are best approximated by the normal distribution, with a mean of 250 minutes and a standard deviation of 20 minutes. How many minutes should she contract for?

For each unused minute at the end of a month, how much does it cost Liz?

How many minutes should she contract for per month to minimize her expected total monthly cost?