Revenue and Cost Functions

Question

A company that manufactures running shoes has a fixed cost of $300,000. Additionally, it costs $30 to produce each pair of shoes. They are sold at $80 a pair.

Write the cost function, C, of producing x pairs of running shoes.
Write the revenue function, R , from the sale of x pairs of running shoes.
Determine the break-even point. Describe what this means.
Students are free to use Microsoft Excel or graph paper.

Answer 1

C(x) = mx + b, where m = variable cost, b = fixed cost, and x = output (# of pairs of shoes).

 

C(x) = 30x + 300,000

 

R(x) = px, where p = price of a pair of shoes and x = output (# of pairs of shoes).

 

R(x) = 80x

 

These two functions will be used to find the profit, P(x).  The profit is the difference of revenue and cost.

 

R(x) - C(x) = P(x)

80x - 30x - 300,000 = P(x)      Plug in the equation for each function except for P(x).

80x - 30x - 300,000 = 0          P(x) = 0 by definition of the break-even point.

50x - 300,000 = 0                   Combine like terms.

50x = 300,000                        Add 300,000 on both sides.

x = 6,000                               Divide 50 on both sides.

 

A company will need to produce 6,000 pairs of shoes in order to have a profit of $0.  This means the revenue and cost will be the same.

 

R(6,000) = 80(6,000) = 480,000

 

C(6,000) = 30(6,000) + 300,000 = 180,000 + 300,000 = 480,000

 

P(6,000) = 480,000 - 480,000 = 0

 

When you graph R(x) and C(x), they will intersect at x = 6,000 pairs of shoes and y = $480,000.