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 obey the demand equation $$x=-20 p+500 \quad 0<p \leq 25$$ (a) Express the revenue $R$ as a function of $x$. (b) What is the revenue if 20 units are sold? (c) What quantity $x$ maximizes revenue? What is the maximum revenue? (d) What price should the company charge to maximize revenue? (c) 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 obey the demand equation $$x=-20 p+500 \quad 0<p \leq 25$$
(a) Express the revenue $R$ as a function of $x$.
(b) What is the revenue if 20 units are sold?
(c) What quantity $x$ maximizes revenue? What is the maximum revenue?
(d) What price should the company charge to maximize revenue?
(c) What price should the company charge to earn at least $\$ 3000$ in revenue?

Key Concepts

Demand Equation

The demand equation represents the relationship between the product’s price and the quantity sold. It conveys how changes in price affect consumer demand and is usually formulated based on market behavior, typically indicating an inverse relationship where an increase in price leads to a decrease in quantity sold.

Revenue Function

The revenue function calculates the total income generated from sales by multiplying the price per unit by the quantity sold. It is a critical concept in business and economics as it allows the assessment of how different pricing or sales levels affect overall earnings.

Quadratic Functions

Quadratic functions are polynomial functions of degree two that graph as parabolas. In the context of revenue problems, the revenue function often turns into a quadratic expression when expressed in terms of quantity or price, and its parabolic nature is useful for finding maximum or minimum values.

Optimization of Revenue

Optimization in revenue problems involves finding the conditions under which revenue reaches its maximum value, which is particularly important for decision-making in business. This is typically achieved by analyzing the quadratic revenue function and determining its vertex, where the maximum revenue occurs.

Function Transformation

Function transformation involves rewriting a function in terms of a different variable. In these types of problems, it is often necessary to express the revenue function solely in terms of quantity sold or price by substituting the demand equation into the revenue formula, thereby facilitating further analysis such as optimization.

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