The monthly profit in dollars for a small consulting firm...

The monthly profit in dollars for a small consulting firm can be modeled by the equation $y=-16.7 x^{3}-400 x^{2}+$ $6,367 x-7,000,$ where $x$ is the number of new clients acquired during that month. For tax purposes, the owner asks the manager to try to make the December profit as close as possible to $\$ 11,000 .$ How many new clients should the manager try bring in?

Key Concepts

Optimization in Models

Optimization in mathematical models focuses on adjusting input values to achieve the best possible output, such as maximizing profit or minimizing error. In scenarios where a target value is desired, the goal is to select the input that results in an output as close as possible to that target, even if an exact match may not be attainable.

Polynomial Equation Solving

This concept involves finding the value(s) of the variable that satisfy a given polynomial equation, particularly when setting the output of a model equal to a specific target value. Techniques for solving such equations can include factoring, using the Rational Root Theorem, or applying numerical methods when an exact solution is difficult to obtain.

Cubic Functions

A cubic function is a type of polynomial function of degree three, characterized by its ability to model complex relationships with potential inflection points and multiple turning points. These functions are used to represent situations where the rate of change itself changes dynamically, a common occurrence in real-world scenarios.

Mathematical Modeling

Mathematical modeling involves representing real-world phenomena with equations or functions. In this context, the profit of a business is modeled as a function of the number of new clients, allowing decision-makers to predict and adjust their actions to achieve desired financial outcomes.

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