Maximizing Revenue The price p (in dollars) and the quantity...

Maximizing Revenue The price $p$ (in dollars) and the quantity $x$ sold of a certain product obey the demand equation $$ p=-\frac{1}{3} x+100 $$ (a) Find a model that expresses the revenue $R$ as a function of $x .$ (b) What is the domain of $R ?$ (c) What is the revenue if 100 units are sold? (d) What quantity $x$ maximizes revenue? What is the maximum revenue? (e) What price should the company charge to maximize revenue?

Maximizing Revenue The price $p$ (in dollars) and the quantity $x$ sold of a certain product obey the demand equation
$$
p=-\frac{1}{3} x+100
$$
(a) Find a model that expresses the revenue $R$ as a function of $x .$
(b) What is the domain of $R ?$
(c) What is the revenue if 100 units are sold?
(d) What quantity $x$ maximizes revenue? What is the maximum revenue?
(e) What price should the company charge to maximize revenue?

Key Concepts

Demand Function

A demand function describes how the price of a product varies with the quantity sold. It encapsulates the relationship between price and the demand or number of units consumers are willing to purchase, typically showing that a higher quantity leads to a lower price per unit, based on market dynamics.

Revenue Function

A revenue function models the income generated from sales as a product of the price per unit and the number of units sold. This function is derived by substituting the demand function into the revenue equation, creating a mathematical model that represents how total revenue changes with respect to quantity sold.

Domain Restrictions

Domain restrictions in this context pertain to identifying all valid values for the quantity sold that make sense in a real-world scenario. For revenue functions in economics, practical constraints such as non-negative sales and price boundaries influence the domain of the function, ensuring that only realistic quantities are considered.

Quadratic Functions and Vertex

When the revenue function is expressed in terms of quantity and leads to a quadratic equation, the graph of the function is a parabola. In cases where the parabola opens downward, the vertex represents the maximum point of the function. Techniques for finding the vertex, such as completing the square or using the formula -b/(2a), are applied to determine the quantity that yields maximum revenue.

Optimization

Optimization in revenue problems involves determining the conditions that maximize or minimize a particular objective function—in this case, revenue. This typically requires setting up a function that models the economic scenario, applying constraints, and using mathematical tools to find the optimal solution where revenue is maximized.

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