Lesson

It's really important for businesses to make a profit! Otherwise, they won't be around for very long. So businesses should know their break-even point, the amount of money they need to take in to cover all their expenses. At this point there is no profit or loss, and their income is equal to their expenses.

To find this point, we use break-even analysis. Break-even analysis looks at the graphs of cost and revenue together to determine where they cross. This point of intersection is the break-even point where income equals expenses.

Here is an example of a break-even analysis for a single day of operation for the company *Lovely Lemonade*.

The horizontal axis is the number of drinks, or units, they sell. The expenses line (red) starts off higher than the income line (green). So for a low number of units sold, *Lovely Lemonade* will lose money. For example, if they only sell $1000$1000 units, the green line tells us they earn $\$200$$200, the red line tells us they spend $\$600$$600, and so overall they make $\$200-\$600=-\$400$$200−$600=−$400. The negative sign means overall they **lose** $\$400$$400.

But eventually, the income line becomes higher than the expenses line, so for a high number of units sold, *Lovely Lemonade* will make money. For example, if they sell $5000$5000 units, the graph tells us that they will earn $\$1100$$1100 but only spend $\$1000$$1000. Overall they make $\$1100-\$1000=\$100$$1100−$1000=$100, and the positive sign means a **profit** of $\$100$$100.

The important point is when the two lines meet - this is the **break-even point**. In this example we can see that selling $4000$4000 units means they don't make any profit, but they don't lose any money either - both their income and their expenses are $\$900$$900. Reaching this amount should be an important first goal for *Lovely Lemonade* every single day!

We can also use break-even analysis to compare two different pricing plans for the same product or service to see which one offers the best deal for a particular situation.

Here's an example comparison between two energy plans provided by rival companies *Thorgate* and *Callisto*:

In this scenario the break-even point at $\left(40,10\right)$(40,10) tells us that both plans charge $\$10$$10 for $40$40 kWh. For energy amounts less than the break-even point we can see that *Thorgate* is cheaper, and for energy amounts more than the break-even point *Callisto* is cheaper. Knowing how much energy someone plans on using can then determine the best plan for them.

Consider the following phone plans:

GO SMALL plan: This plan has a $\$20$$20 monthly base charge and charges $90$90 cents per minute for all calls.

GO MEDIUM plan: This plan has a $\$26$$26 monthly base charge and then charges $70$70 cents per minute for all calls.

Complete the following table of values for various total monthly call times for the two plans:

Call time (in minutes) Total cost for GO SMALL plan Total cost for GO MEDIUM plan $20$20 $\editable{}$ $\editable{}$ $30$30 $\editable{}$ $\editable{}$ $40$40 $\editable{}$ $\editable{}$ $50$50 $\editable{}$ $\editable{}$ Sketch the graph of the two plans.

Loading Graph...Using the graphs, determine how many minutes of calls would need to be made so that the monthly bill costs the same on both plans.

This graph shows the cost $C\left(x\right)$`C`(`x`), the revenue $R\left(x\right)$`R`(`x`) and the profit $P\left(x\right)$`P`(`x`) from making and selling $x$`x` units of a certain good.

Loading Graph...

Using the labelling of the lines, identify which line corresponds to which function.

$P\left(x\right)$

`P`(`x`) $=$= $\editable{}$$C\left(x\right)$

`C`(`x`) $=$= $\editable{}$$R\left(x\right)$

`R`(`x`) $=$= $\editable{}$How many units must be sold to reach the break even point?

Use curve fitting, log modelling, and linear programming techniques

Apply linear programming methods in solving problems