Exercise 2.
Given \(\Dprice=-2 \quantity+20\) and \(\cost=3 \quantity+10\text{,}\) with \(q_0=6\text{.}\)
Solution.
  1. Identify the fixed and variable costs.
    The fixed cost is $10 (the constant/fixed part of the cost function), and the variable cost is $3 per item.
  2. Find the revenue and profit functions
    \begin{align*} \revenue\amp=\Dprice*\quantity\\ \amp=(-2 q+20)*q=-2 q^2 +20 q \end{align*}
    \begin{align*} \profit= \revenue-\cost \amp=-2 q^2 +20q-(3q+10)\\ \amp=-2 q^2+17q-10\text{.} \end{align*}
  3. Evaluate cost, demand price, revenue, and profit at \(q_0\)
    \begin{align*} \cost(6)\amp =3 (6)+10=28\\ \Dprice(6)\amp =-2 (6)+20=8\\ \revenue(6)\amp =-2 6^2 +20 (6)=-72+120=48\\ \profit\amp =\revenue-\cost =48-28=20\text{.} \end{align*}
  4. Find all break-even points
    Solve \(\profit=-2 q^2+17q-10=0\text{.}\) We can do this with Excel or with Wolfram. The break even points are \(q = 0.6\) and \(q = 7.9\text{.}\)
  5. Graph the profit function over a domain that includes both break-even points. Add a textbox and label to identify the first break-even point.
    If we had done the whole problem in Excel it would look like this:
    Entries in the cells before quick fill
    The table with \(q\) between 0 and 8
    Goal Seek gives break even points at \(q = 0.64\) and \(q = 7.86\)
    The graph produced in Excel. To create the labels: double click on one of the break-even points, go to “Chart Layout”, go to “Data Labels”. In this example we chose the “x-value”. Both break-even points were labeled to show where they are and what the values are in this problem.
in-context