Cost, Revenue, Profit and Taxes
Identify the relevant information given to you in the application problem below. Use that information to answer the questions that follow on cost, revenue and profit.

Round your answers to two decimal places as needed.
You decide to begin selling bottles of water at the local art festival. Your cost for each bottle of water is $3.00 plus you have to pay a fixed weekly fee of $170 for the booth. Your plan is to sell each bottle of water for $4.00.

To help with planning, you come up with the following functions.
  1. The total costs for the week is $3.00 per item plus the weekly rent of $170.
    C(n)=3.00n+170\displaystyle {C}{\left({n}\right)}={3.00}{n}+{170}
  2. The amount of revenue for the week is $4 times the number of items sold.
    R(n)=4n\displaystyle {R}{\left({n}\right)}={4}{n}
  3. Your profit for the week is the amount of revenue minus the cost of business.
    P(n)=R(n)−C(n)=1n−170\displaystyle {P}{\left({n}\right)}={R}{\left({n}\right)}-{C}{\left({n}\right)}={1}{n}-{170}
Use the function for profit P(n)\displaystyle {P}{\left({n}\right)} to complete the table below. This will give you an idea of how much you can earn selling n\displaystyle {n} bottles of water.

n5001000150020002500
P(n)
This should be looking pretty good. But the owners of the art festival inform you that you will have to pay a "self employed income tax" on your earnings of 12%. When you take this into account, you determine your Income is actually your Profit - 12% of your Profit or a better way to state it is that your Income is 88% of your profit. This can be written as the function

I(p)=0.88p\displaystyle {I}{\left({p}\right)}={0.88}{p}
Use the function for Income I(p)\displaystyle {I}{\left({p}\right)} to complete the table below. This will give you an idea of how much your income is if your profit is p\displaystyle {p}

p330830133018302330
I(p)
After taking everything into consideration, you decide you need to have a function that tells you your Income if you sell n\displaystyle {n} bottles of water. Your Profit you make is $1 per bottle of water sold minus $170. Your Income is 88% of your Profit. So your function is:

I(P(n))=0.88⋅P(n)=0.88(1n−170)\displaystyle {I}{\left({P}{\left({n}\right)}\right)}={0.88}\cdot{P}{\left({n}\right)}={0.88}{\left({1}{n}-{170}\right)}
Use your function for Income I(P(n))\displaystyle {I}{\left({P}{\left({n}\right)}\right)} to complete the table below. This will give you an idea of how much your income is if you sell n\displaystyle {n} bottles of water.

n5001000150020002500
I(n)

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