The monthly revenue R for a product is given by R-420 x-2...

The monthly revenue $R$ for a product is given by $R-420 x-2 x^{2},$ where $x$ is the price in dollars of each unit produced. Find the interval in terms of $x$ for which the monthly revenue is greater than zero.

Key Concepts

Revenue Modeling

Revenue modeling uses mathematical functions to represent income in terms of variables such as price or quantity. In many cases, revenue functions are quadratic, reflecting diminishing returns at higher prices or production levels. Analyzing these models helps businesses determine the optimal price range to ensure profitability.

Quadratic Functions

Quadratic functions are polynomial functions of degree two and are generally expressed in the form ax² + bx + c. They are key in understanding various real-world phenomena such as revenue models, where the function's shape (a parabola) can indicate maximum or minimum values.

Factoring

Factoring is a method used to simplify polynomial expressions by expressing them as a product of their factors. It is particularly useful in solving equations and inequalities, as it allows one to identify the roots or zeros of the function, which are critical for determining where the function is positive or negative.

Solving Inequalities

Solving inequalities involves finding the set of values for the variable that make the inequality true. For quadratic inequalities, this typically includes factoring the quadratic to find its critical points, and then using sign analysis to determine the intervals where the inequality holds.

Transcript

00:01 So in this problem, we're told that the equation r equals 420x minus 2x squared is going to represent the revenue for some company.

00:10 So what we're being asked to find is the interval in terms of x for which the monthly revenue is greater than zero.

00:17 Meaning we need to figure out when is this expression greater than zero.

00:21 So we have 420x minus 2x squared greater than zero.

00:26 Okay.

00:26 Well, as you can see, we have a quadratic inequality.

00:30 So the first thing we have to do is find the zeros, meaning we have to set our quadratic equal to zero.

00:35 So 420, excuse me, x minus 2x squared would equal to zero.

00:41 So i'm going to solve this by factoring.

00:43 So i'm going to factor out the greatest common factor, which is going to be 2x.

00:48 And then we'll divide both terms by 2x.

00:50 Well, 420x divided by 2x is equal to 210, and negative 2x squared divided.

00:57 By 2x is negative x.

01:00 Okay, great.

01:00 So now that we've factored, we'll set both of our factors equal to 0.

01:04 So 2x could equal to 0 or 210 minus x could equal to 0.

01:09 So to solve the first equation, we'll divide both sides by 2, so x will be 0.

01:13 And to solve the second equation, i'm just going to add x to both sides because then we'll have x is equal to 210.

01:20 Perfect.

01:21 So now we've found our two zeros...

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