The monthly profit function for a product is given...

The monthly profit function for a product is given by$$P(x)=-x^{3}+10 x^{2}$$ where $x$ is the number of units sold measured in thousands and $P$ is profit measured in thousands of dollars. The average profit, which represents the profit per thousand units sold, for this product is given by$$P(x)=\frac{-x^{3}+10 x^{2}}{x}$$ where $x$ is units sold measured in thousands and $P$ is profit measured in thousands of dollars. Find the number of units that must be sold to produce an average profit of $$ 16,000$ per thousand units. Convert the answer to average profit in dollars per unit.

The monthly profit function for a product is given by$$P(x)=-x^{3}+10 x^{2}$$ where $x$ is the number of units sold measured in thousands and $P$ is profit measured in thousands of dollars. The average profit, which represents the profit per thousand units sold, for this product is given by$$P(x)=\frac{-x^{3}+10 x^{2}}{x}$$ where $x$ is units sold measured in thousands and $P$ is profit measured in thousands of dollars.
Find the number of units that must be sold to produce an average profit of $$ 16,000$ per thousand units. Convert the answer to average profit in dollars per unit.

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Key Concepts

Unit Conversion and Scaling

In many real-world problems, variables are measured in units like thousands. Understanding how to convert these values—such as translating profit per thousand units into profit per single unit—is essential to accurately interpret and apply the results.

Polynomial Functions

Polynomial functions, such as the given cubic and its derived quadratic form, are commonly used in economic modeling. They capture the nonlinear relationships between variables like sales and profit, helping to predict overall behavior.

Quadratic Equations

The process of solving quadratic equations, whether by factoring, completing the square, or using the quadratic formula, is a fundamental algebraic skill. It allows one to find specific values (such as sales levels) that satisfy the desired condition in a modeled scenario.

Profit Function

A profit function models the total profit obtained from selling a certain number of units. It is usually expressed as a polynomial, which helps in understanding how profit changes with variations in sales volume.

Average Profit

Average profit represents the profit earned per unit, derived by dividing the total profit function by the number of units sold. This concept is important for assessing the efficiency and performance at different scales of production.

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