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 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.
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 best approximated by the Normal distribution, with a mean of 250 minutes and a standard deviation of 24
minutes. [Hint: You might think about the problem this way: if you pick too many minutes then you have “left over inventory” but if you pick too few minutes then you have “lost sales”.]
If Marisol chooses the “Pick Your Minutes” plan described above, how many minutes should she contract for?
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