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:

Inductive Reasoning

A method of reasoning in which specific observations or experiences are used to make broad generalizations.

Unwarranted Assumption

A belief taken for granted without sufficient evidence or basis, leading to potential misunderstandings or erroneous conclusions.

Deductive Reasoning

A logical process in which a conclusion is based on the concordance of multiple premises that are generally assumed to be true.

Creative Problem-Solving

An approach to finding innovative solutions to complex problems through creativity and non-standard strategies.