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
$$
R(p)=-4 p^{2}+4000 p
$$
(a) At what prices $p$ is revenue zero?
(b) For what range of prices will revenue exceed $\$ 800,000 ?$

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve finding the range of values for the variable for which the quadratic function satisfies an inequality, such as being greater than or less than a certain number. Solving these inequalities usually starts by finding the zeros of the corresponding quadratic equation and then testing intervals. This approach is crucial when determining conditions under which certain criteria, like exceeding a revenue threshold, are met.

Revenue Modeling

Revenue modeling applies mathematical functions to represent and analyze the relationship between unit price and total revenue. Often modeled by quadratic functions, these models help in understanding how changes in price can impact overall revenue, and they provide valuable insights for decision-making in business contexts.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form ax² + bx + c. Their graphs are parabolas that open upward if a is positive and downward if a is negative. The properties of quadratic functions, such as their vertex, axis of symmetry, and intercepts, are useful in various real-world applications, including revenue modeling and optimization problems.

Zeros of a Function

The zeros of a function (or roots) are the values of the variable for which the function equals zero. In the context of quadratic functions, finding the zeros often involves factoring, completing the square, or applying the quadratic formula. Identifying these zeros is important for understanding when a process, such as generating revenue, begins to yield no returns.

Transcript

00:01 This question is a revenue question revolving around a company and their prices.

00:07 And they find if they overpriced too much, they will start to lose money.

00:13 And so what they want to know is at what point is their revenue equal to zero? this equation right here represents their revenue for their product for the corresponding price p.

00:30 And so the question a here is asking at what price value will the revenue equals zero.

00:39 And what that means is in terms of a parabola, in terms of a coordinate plane, we are asking, when does the parabola touch the x -axis? well, we can take a look over here with this graph, and you'll see that this parabola, negative 4p squared minus 4 ,000, touches the x -axis.

01:00 At two locations zero -zero, which makes sense because at $0, they would make zero money.

01:06 They increase their value, their revenue as they increase the price, but then they eventually get to another point when their revenue is $0.

01:16 So their revenue is $0 .00.

01:18 How much is the price of their product? a thousand dollars.

01:24 So here we are going to say p is equal to at p equal to a thousand and that's in dollars a thousand dollars is when the revenue is equal to zero question b is asking for what range is the revenue what price range is the revenue greater than eight thousand dollars eight hundred thousand dollars so again using my graph and calculator desmos i have here eight hundred thousand dollars and it's in between these two values...

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