Maximizing Revenue Suppose that the manufacturer of a gas...

Maximizing Revenue Suppose that the manufacturer of a gas clothes dryer has found that when the unit price is $p$ dollars, the revenue $R$ (in dollars) is
$$
R(p)=-4 p^{2}+4000 p
$$
What unit price $p$ maximizes revenue? What is the maximum revenue?

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Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, usually expressed in the general form ax² + bx + c. They are widely used in modeling various real-world scenarios, including revenue and profit, where the relationship between variables is parabolic in nature. This concept is crucial for understanding how to optimize a function since many practical applications result in quadratic models.

Vertex of a Parabola

The vertex of a parabola represents the maximum or minimum point of a quadratic function. When a quadratic function opens downward (i.e., the coefficient of the quadratic term is negative), the vertex yields the maximum value of the function. The vertex can be found using the formula -b/(2a), which is a key step in solving optimization problems involving quadratic functions.

Revenue Optimization

Revenue optimization involves determining the conditions under which a revenue function reaches its maximum or minimum value. In many economic models, revenue is expressed as a quadratic function of price or quantity. By analyzing the vertex of the quadratic function, one can identify the price that maximizes revenue and calculate the maximum potential revenue, making this concept particularly important in business and economics.

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