Marisol is new to town and is in the market for cellular phone service. She has settled on Wildcat Cellular, which will give her a free phone if she signs a one-year contract. Wildcat offers several calling plans. One plan that she is considering is called "Pick Your Minutes". Under this plan, she would pay a fixed fee of $5.00 per month; in addition, she would also specify a quantity of minutes, say x, per month that she would buy at 5¢ per minute. Hence, her upfront cost would be $5 + $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 minutes per month and her actual usage is 40 minutes, her total bill is $5.00 + 120 ´ 0.05 = $11.00. However, if actual usage is 130 minutes, her total bill will be $5.00 + 120 ´ 0.05 + (130 - 120) ´ 0.40 = $15.00.
The same rates apply whether the call is local or long distance. Once she signs the contract, she cannot change the number of minutes specified for a year. Marisol estimates that her monthly needs are described by a normal distribution. This distribution has a mean of 250 minutes and a standard deviation of 24 minutes. If Marisol chooses the "Pick Your Minutes" plan described above, how many minutes should she contract for?