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 obcy 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?

The price $p$ (in dollars) and the quantity $x$ sold of a certain product obcy 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

Optimization Using the Vertex of a Parabola

The vertex of a quadratic function is the point where the maximum or minimum value occurs. In revenue optimization problems, finding the vertex of the revenue function (when the function is a downward-opening parabola) allows one to determine the quantity that yields the highest possible revenue, as well as the corresponding maximum revenue.

Quadratic Functions

When a linear demand function is used to generate a revenue model by multiplying price by quantity, the resulting function is quadratic. Quadratic functions are characterized by a parabolic shape, which can open upwards or downwards, indicating whether they have a minimum or maximum value.

Domain Restrictions

In mathematical modeling, particularly in economic applications, the domain of a function is restricted to values that make sense in a real-world context. For instance, in revenue functions, the quantity sold must be nonnegative and often has an upper limit based on market demand or production capacity.

Revenue Modeling

Revenue is modeled by multiplying the unit price by the quantity sold. This concept is fundamental in economics and business as it quantifies the total income generated and helps in analyzing the effect of varying production levels and pricing strategies.

Demand Function

A demand function expresses the relationship between the product’s price and the quantity sold, typically capturing consumer behavior in response to changes in price. It is often modeled as a linear equation in basic economic scenarios, showing how price decreases as quantity increases.

Transcript

00:01 So they give us an equation for the price or an object, and x is the number sold.

00:15 And then we have to find an equation for the revenue.

00:19 So the revenue in terms of x, well, we would take how many are sold times what the price is, which means we'd have x times negative one -third x plus 100.

00:32 Now that's fine in that factored form, but one thing i want you to notice, is if i distribute that, this does form a quadratic.

00:42 And because the lead term is negative, it's going to be a parabola that opens downward.

00:48 And i can also see where the zeros are for this parabola.

00:52 And that's going to be helpful to us to answer the rest of the questions.

00:56 So i think when does this equal zero? well, when x is zero, and then when does negative 1 3rd, 1 3rd x plus 100, when does that equal 0? and that's when 100 is equal to one -third x and multiplying both sides by three.

01:15 I get that's when x is 300.

01:17 So my zeros to this quadratic are at zero and at 300.

01:24 And i know the problem is a problem that opens downward like that.

01:29 And i also know it's symmetrical.

01:31 So i know the middle here is at 150.

01:34 So i know so, so much without answering even the rest of the questions.

01:39 So part b asking, what should i use for the domain of r? and i can see from my picture, well, it's silly to use anything that isn't from zero up to 300.

01:53 And in both of those, you really don't even want zero or 300 because you make no revenue.

01:58 So kind of between would even make more sense.

02:01 Then we had to find what's the revenue if we make 100...

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