If you are looking for where the break-even points are, you must determine the quantity for which
revenue = cost
Alternately, you may find the profit by calculating
profit = revenue – cost
The problem below demonstrate these strategies starting from a demand function and a cost function. To apply either of the relationships above, you need to form the revenue function from
revenue = (price)(quantity)
where the price is given by the demand function and Q represents the quantity.
Problem The demand function for Q units of a product is given by
The cost function is given by the function
a. Find the revenue function R(Q).
b. Find the break-even point(s)?
c. On a graph of R(Q) and C(Q), where do the break-even points lie?
d. Find the profit function P(Q).
e. Where do the break-even points lie on the graph of P(Q)?
Solution 1 To find the break-even point, this group of students set R(Q) = C(Q). This results in a quadratic equation. They moved all terms to one side and used the quadratic formula to find the quantities at which the revenue is equal to the cost.
Solution 2 This group of students found the profit function P(Q) first. Then they set it equal to zero to find the break-even points. Like the first solution, they also needed to use the quadratic formula.
Both techniques lead to the same break-even points and are equally valid. The only thing the second solution left out was the graph of the profit function showing the break-even points at the zeros (horizontal intercepts) of the function.