Expert: Josh - 11/26/2007

Question
A company that manufactures bicycles has a fixed cost of $100,000. It costs $100 to produce each bicycle. The selling price per bike is $300.

1. Write the cost function, C.

2. Write the revenue function, R.

3. Determine the break-even point. Describe what this means.

Answer
Dean,

The cost function consists of two components. A fixed cost (let's call this C1) and a per-unit manufacturing cost (C2). So the total cost depends on the number of bikes the company makes. We can write C=C1+C2*n, where n is the number of bicycles produced. To show explicitly the dependence of the cost on the number of bicycles produced, we can write C(n)=C1+C2*n, where C1=100,000 and C2=100. This is pretty much standard notation, its interpretation is the same as the function f(x)=m*x+b which represents a straight line; except the variable here is given by "n" instead of "x".

Part 2: Revenue is the money you make from selling the bikes, S(n), less the total cost, C(n).

R(n)=S(n)-C(n), where S(n)=300*n

R(n)=300*n-[100000+100*n] simplifies to
R(n)=200*n-1000000

Part 3: Break-even point is where R(n)=0. The money made from selling the bikes is just sufficient to cover the cost of producing them. i.e., net profit is zero.

Solving R(n)=200*n-1000000=0 for "n", we get n=1000000/200, or 5000 units. Check this yourself just in case I copied the numbers wrong.