FAQ Topic: Business Applications of Extrema

How Do You Find Break-Even Points?

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

$latex \displaystyle D\left( Q \right)=16-1.25Q$

The cost function is given by the function

$latex \displaystyle C\left( Q \right)=2Q+15$

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.

How Do You Find the Economic Order Quantity?

Although many textbooks use a restrictive formula to find economic order quantity or economic lots size, you can use tables to come up with more general formulas.

Here are more examples from class.

Problem 1 A restaurant has an annual demand for 1200 bottle of wine. It costs $1 to store one bottle for a year and $5 to place an order. Orders are made when the inventory of wine is zero. If each bottle costs $15, find the optimum number of bottles per order.

The strategy above is correct, but there is a math mistake in the last board…do you see their error?

Problem 2 If the restaurant in Problem 3 orders wine when the inventory is half of an order size, find the optimum number of bottles per order.

This problem can’t be done with the formula given in Help Me Solve This. The change in when the order is made (when the inventory drops to half the order size) leads to a different expression for the storage costs. However, once this pattern is established the cost function can be maximized as before.