Mr. Hershman is a busy, busy person (at least he thinks he is). He has to divide his time after school between sleeping and taking care of his kids, cooking, cleaning, and tending to the home front, spending time with his wife, keeping up with his friends, taking pictures at the climbing gym, and writing math problems – let’s just call it _____ (choose a description of Mr. Hershman's non-sleeping activities to fill in this and *all *remaining blanks).

Determining how to balance his time between _____ and sleeping is a difficult task and he needs your help!

To begin with Mr. Hershman has a basic allotment of time *each day* he spends outside of school which is roughly from 4pm until 8am, so his total _____ and sleeping time falls within this interval. Each hour spent _____ he accomplishes 5 units worth of work and he decides to place a value of 3 work units on each hour spent sleeping. He must not let his total number of __work units__ fall below 30 for any day because then he would feel “behind” and that’s not good.

It’s also important for Mr. Hershman to feel __rested__, he decides that time spent _____ can be restful in the sense that productivity can create the space for relaxation, so he also decides to count each hour _____ as 1 restfulness and each hour spent sleeping as 3. He’s a busy person though so he can’t accumulate more than 34 restful points in any day without feeling lazy. And, yup, laziness is no good either.

As if being productive and restful weren’t all, it is important that he maintain some ability to converse with other humans. Being awake / doing the things that encompass _____ yield roughly 5 topics of possible __conversation__ per hour. While each hour of blissful sleep Mr. Hershman snoozes causes him to forget 3 topics. Caution is important though, too many topics of conversation can make it difficult to discuss anyone in particular, and so he chooses 40 as his arbitrary limit for the number of topics of conversation that he should accumulate in any single day.

So, in the end, Mr. Hershman wants to balance his total hours between ______ and sleeping. He must pay attention to his productivity, restfulness, and conversation. Ultimately he believes that each hour ______ should be given a weight of 3 happiness units and each hour spent sleeping should be given a weight of 5 happiness. How many hours should he spend sleeping and how many hours should he spend _____ in order to maximize his happiness???

*Instructions**: Define two variables, write a set of inequalities to represent the constraints, plot the inequalities carefully on the graph, identify the feasible region and determine the point of maximum happiness. Can you think of a real life dilemma faced by a friend or family member – a situation in which two or more options must be put in balance? Create your own optimization problem*.

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This problem is tonight's homework.