   SEARCH HOME Math Central Quandaries & Queries  Question from virginia, a parent: a company that manufactures bikes has a fixed cost of $100,000.00 it costs$100 to produce each bike. the selling price per bike is $300.00 write the cost function, c write the revenue function, r determine the break even point im new to all this type of math and i dont understand it to explain it to my kids. We have two responses for you Hi Virginia. The "Cost" is how much money it takes to produce a given number of bikes. Say you want to produce 2000 bikes, then the cost is$100 000 + $100 x 2000 =$300 000. The cost function uses a variable in place of the number of bicycles. So the cost "c" is based on the number of bikes "b":

c = $100 000 +$100 x b.

The "Revenue" is the amount of money the company takes in due to selling these bikes. The revenue function works the same way:

r = $300 x b. The break even point, in any business, is where your revenue (income) is just enough to pay your costs (expenses). This means c = r. Remember that the equal sign means that what is on the right side is interchangeable with what is on the left side, so we can start with c = r, and then replace the c with what c equals, and the same for r. Thus, 100 000 + 100 b = 300 b. The break even point is the number of bicycles the company must sell to meet its expenses with its sales. Solve for b. Hope this helps, Stephen La Rocque. Hi Virginia, A function is a mathematical way of describing a real life relationship between certain things. In your problem, the "cost function" would be a mathematical expression that details how the company would compute their cost of manufacturing bikes. Since the number of bikes they produce may vary from year to year, a variable would be used in this function to represent the unknown number of bikes produced. Since the cost of manufacturing bikes depends on the number of bikes produced, cost also needs a variable. Let's use C to represent the cost and b to represent the number of bikes produced. According to your problem, regardless of the number of bikes produced, the company incurs costs of$100,000. It also costs them $100 per bike they manufacture. Therefore the cost function would look as follows: C = 100b + 100,000 In words "their cost is$100 per bike plus $100,000". With this function they could determine their cost once they know the number of bikes they produce. Their revenue function is even simpler: "revenue is$300 per bike"
Let's use R for revenue and, again, b for the number of bikes.
R = 300b
They can determine their revenue once they know the number of bikes sold.

In business, "breaking even" means that costs equal revenues, that is, the company neither makes a profit, nor takes a loss. The "break-even point" for this bike manufacturer would have to represent the exact number of bikes needed in order for their cost, C, to equal their revenue, R.

In other words, what value must b take on in order for 100b + 100,000 = 300b?

Hope this helps,
Leeanne     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.