The price p (in dollars) and the quantity x sold of a...

The price $p$ (in dollars) and the quantity $x$ sold of a certain product satisfy the demand equation $$ x=-20 p+500 $$ (a) Find a model that expresses the revenue $R$ as a function of $p$. (b) What is the domain of $R ?$ Assume $R$ is nonnegative. (c) What price $p$ maximizes the revenue? (d) What is the maximum revenue? (e) How many units are sold at this price? (f) Graph $R$. (g) What price should the company charge to earn at least $\$ 3000$ in revenue?

The price $p$ (in dollars) and the quantity $x$ sold of a certain product satisfy the demand equation
$$
x=-20 p+500
$$
(a) Find a model that expresses the revenue $R$ as a function of $p$.
(b) What is the domain of $R ?$ Assume $R$ is nonnegative.
(c) What price $p$ maximizes the revenue?
(d) What is the maximum revenue?
(e) How many units are sold at this price?
(f) Graph $R$.
(g) What price should the company charge to earn at least $\$ 3000$ in revenue?

Key Concepts

Inequality Analysis in Revenue Context

Inequality analysis is applied when there are conditions, such as achieving a minimum revenue target. This involves setting up inequalities based on the revenue function and solving for the input variables that satisfy these conditions. It ensures that the business objectives, like a minimum revenue threshold, are met within the feasible domain.

Graphical Analysis of Functions

Graphical analysis involves plotting the function on a coordinate system to visualize its behavior. In economic problems, graphing a revenue function can help illustrate how revenue responds to changes in price, identify optimal pricing strategies, and understand the impact of constraints such as minimum revenue requirements.

Optimization of a Quadratic Function

Optimization involves finding the maximum or minimum values of a function, which is a common task in revenue analysis. For quadratic functions, the vertex of the parabola represents the optimal point, such as the maximum revenue in a revenue model. Techniques such as completing the square or using the vertex formula are typically employed.

Revenue Function Modeling

A revenue function expresses the revenue generated as a function of price, quantity, or both. It is usually derived by multiplying the price at which a product is sold by the quantity sold. This model is used to analyze how revenue varies as either price or demand changes, providing a framework for business decision-making.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, usually having the form ax² + bx + c. In the context of revenue functions, when the linear demand equation is used in the revenue model, the resulting function is quadratic. These functions are characterized by their parabolic shape, whereby the maximum or minimum value (vertex) can be calculated directly.

Demand Equation

A demand equation represents the relationship between the price of a product and the quantity of the product that consumers are willing to purchase. Typically, this is modeled as a linear function in basic economic problems, but it can also take on other forms in more complex scenarios. The equation helps in understanding how a change in price affects the quantity demanded.

Domain of a Function

The domain of a function describes the set of input values for which the function is defined. In revenue modeling, it is crucial to consider realistic constraints such as nonnegative revenue and nonnegative quantities. Determining the domain ensures that the model only considers economically feasible scenarios.

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