# Project Help!!!!!Research a local business and find out what you can about their profit and costs. After doing so complete the following tasks: Create 2 linear functions, 1 to model the cost of the business and the other to model the revenue of the business (3 points for each funtion). The template for you model might look something like: revenue = (amount made off each customer)(number of customers) cost = (cost created by each customer)(number of customers) + (some flat cost incurred ) cost = some flat cost incurred Using your work from 1) and the relationship between revenue, cost and profit, find the linear relation that you would expect to model profit. Any help would be great! To accomplish these kinds of projects, you should first ask a local shop/business (even just the small ones) and ask permission to either monitor their business for a couple of weeks (20 days?) or permission if they would show you their books for say, a month. Alternatively, as most probably won't agree to do so, you can just interview the owner (though it's better to have actual data points).

First, you need to develop a revenue function, I(x), where x is the number of days of operation. Most probably, it would be of the form:

`I(x) = ax + b` where a is the average revenue per day, and b is any exogenous income they get that is usually fixed (either through tips, or rents)

Then, you need a cost function, C(x, y), where x is the number of customers, and y is the days of operation. The dependence of the function on x is associated with an approximate cost, a, that the business incurs, if any, for every customer, while its dependence on y can be seen as the cost of operation, like electricity or water.

`C(x, y) = ax + by + c` while c, on the other hand, will be any fixed cost (like rent, if they do not own the place, or taxes, though taxes can be modelled separately as another function).

This would be your two linear functions (you can drop the by on the cost function so that you only need a two dimensional graph).

The profit function then would simply be the difference between the two.

`Pi(x) = I(x) - C(x)`

If you plot all the functions, the point where I and C intersect would be the break-even point (profit = 0). The slope of the profit curve will also show you how successful the business is. If it's negative, cost is greater than revenue, and the business is failing.

These are all linear functions, so most analysis would fall on analyzing the slopes, and the intersection of the lines.

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