A Business Consultant Analyzing a Small Computer Company Finds That $x$

question 33

Essay

A business consultant analyzing a small computer company finds that if the company produces and sells $x$ microprocessors it has a total profit function (in thousands of dollars) given by $p ( x ) = 0.4 x ^ { 2 } - 0.0043 x ^ { 3 } + 7 x - 244$ , with $x$ (in hundreds), being the number of processors produced and sold annually.
(a) Graph the profit function and determine how many processors must be produced and sold in order to break even.
(b) What is the domain of the profit function where the company actually loses money?
(c) Estimate the number of processors that must be produced in order to give the company the largest possible profit.
(d) Estimate the largest profit.

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Definitions:

Collude

To come to a secret understanding for a harmful purpose; often refers to competitors agreeing on prices or market shares to gain an advantage.

Annual Fixed Cost

Regular expenses incurred by a business or individual that do not vary with production level or output within a year.

Marginal Cost Curve

A graphical representation showing the change in total cost that comes with producing one more unit of an item.

Marginal Revenue Curve

A graph showing how marginal revenue varies as output quantity changes, important for understanding profit maximization.