Need help with cost function question

Question

Scenario:

Global Furniture is a company that manufactures office chairs.The market research department of the company has presented for the daily production of x office chairs,the following price-demand function, p =250−1.5x, and cost function, C =600+15x. As part of the market research department, analyse the revenue function and profit function for the production of x office chairs, while identifying what the marginal revenue and marginal profit signifies. Also, determine the exact and approximate revenue from the manufacturing of the 25th office chair on any particular day.

Answer 1

Hi Zyniere

One step at a time:

1) The price/demand function p(x) = 250-1.5x defines the price charged as a function of the number of chairs sold in a day.

a) Revenue is price times volume, so the revenue function is p(x) times x, or R(x) = 250x - 1.5x^2

b) Profit is Revenue minus Cost - let's call it Margin, so M(x) = R(x) - C(x)

Not sure what they mean by "analyze" these functions, but you note that both Revenue and Margin are quadratics. This means you can find the optimum revenue by setting the derivative of R(x) = 0. Solving for x gives you the number of chairs per day that maximizes the revenue. The Margin or profit is a different function, so setting the derivative of M(x) = 0 and solving for x find the number of chairs per day that maximizes profit. Note that those numbers are likely to be different from each other.

2) The marginal revenue and marginal profit measures the change in each as the number of chairs/day changes. They are calculated by taking the derivative of each function and evaluating it at the specific number of chairs you are interested in. Thus:

a} The exact marginal revenue created by the 25th chair in a day is the derivative of R(x) evaluated at 25

or R'(25)

b) The approximate additional revenue can be estimated by finding R(24) and R(25) (note: not R'), and calculating the difference, i.e., R(25) - R(24). You're subtracting the revenue for selling 25 chairs from the revenue for selling 24 chairs. It should be close to, but not exactly equal to R'(25).

Hope that helps.

Jim