Suppose that the manufacturer of a gas clothes dryer has...

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 should be established for the dryer to maximize revenue? What is the maximum revenue?

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Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, generally expressed in the form ax² + bx + c. The graph of a quadratic function is a parabola, and the sign of the coefficient a determines if the parabola opens upwards (minimum point) or downwards (maximum point). This understanding is essential when modeling and solving optimization problems.

Vertex of a Parabola

The vertex of a parabola is the point at which the function reaches its maximum or minimum value, depending on the orientation of the parabola. When the quadratic coefficient is negative, the vertex represents the maximum value of the function. The x-coordinate of the vertex can be found using the formula -b/(2a), which is a critical step in optimizing quadratic models.

Revenue Optimization

Revenue optimization involves determining the price or quantity that yields the highest revenue. When revenue is modeled by a quadratic function, using the vertex of the parabola enables one to identify the optimal pricing strategy that maximizes revenue, making this concept crucial for practical decision-making in economics and business.

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