A market analyst working for a small-appliance manufacturer...

A market analyst working for a small-appliance manufacturer finds that if the firm produces and sells $x$ blenders annually, the total profit (in dollars) is $$P(x)=8 x+0.3 x^{2}-0.0013 x^{3}-372$$ Graph the function $P$ in an appropriate viewing rectangle and use the graph to answer the following questions. a. When just a few blenders are manufactured, the firm loses money (profit is negative). (For example, $P(10)=-263.3,$ so the firm loses $\$ 263.30$ if it produces and sells only $10 \text { blenders. })$ How many blenders must the firm produce to break even? b. Does profit increase indefinitely as more blenders are produced and sold? If not, what is the largest possible profit the firm could have?

A market analyst working for a small-appliance manufacturer finds that if the firm produces and sells $x$ blenders annually, the total profit (in dollars) is $$P(x)=8 x+0.3 x^{2}-0.0013 x^{3}-372$$ Graph the function $P$ in an appropriate viewing rectangle and use the graph to answer the following questions.
a. When just a few blenders are manufactured, the firm loses money (profit is negative). (For example, $P(10)=-263.3,$ so the firm loses $\$ 263.30$ if it produces and sells only $10 \text { blenders. })$ How many blenders must the firm produce to break even?
b. Does profit increase indefinitely as more blenders are produced and sold? If not, what is the largest possible profit the firm could have?

Key Concepts

Optimization

Optimization in economics and business refers to the process of finding the best solution under given constraints, such as determining the production level that maximizes profit. It often involves analyzing turning points and the behavior of functions to ensure that the maximum or minimum value is identified accurately.

Graphing Functions

Graphing functions involves plotting a function on a coordinate plane within a suitable viewing rectangle to visually interpret the behavior of the function. This process aids in understanding the relationship between variables, identifying intercepts, extrema, and overall trends, which is particularly useful in decision-making scenarios.

Cubic Functions

Cubic functions are polynomial functions of degree three that often exhibit characteristic turning points and inflection points. They can model complex relationships in business contexts, such as profit behavior over a range of production levels, and are analyzed to understand patterns such as periods of increasing profit, maximum profit, and losses.

Profit Function

A profit function expresses the firm’s profit as a function of the number of units produced and sold. It is used to determine how changes in production levels affect profitability and to analyze scenarios such as break-even points, maximum profit, and losses.

Break-even Analysis

Break-even analysis involves finding the production level at which total revenue equals total costs, resulting in zero profit. It is a critical concept in economics and business that helps determine the minimum output needed for a firm to avoid losses and start making a profit.

Transcript

00:02 So in this question i have a px function given to me as this expression where p represents the total profit made by a company for selling x number of blenders so i'll extend this coordinate axis this is the x axis and this is the y axis so if i were to draw the graph for this first of all i have to tell the roots so let us see the roots of this two big expression the roots are x equal to minus minus 45 .22 and 25 .233 and one more route will be coming that will come out as 250.

00:46 So i am marking this routes randomly on the coordinate axis.

00:51 Minus 45 is somewhere over here, 25 is somewhere here let me say and 250 is somewhere here, let's call it.

01:01 Now it is to be observed that the leading coefficient is negative over here i can see this.

01:10 So i can see that the leading coefficient is negative so the graph will be decreasing from minus negative.

01:17 So it will come down like this.

01:20 Keep going down down down and cut somewhere the y -axis then again increase this way.

01:25 Will get some higher values then again cut the graph at the x -axis at the point 250 .0 so let me mark these points on this minus 25 .22 .0 next this point is 25 .233 .0 and the last point is 250 .76 .0.

01:52 So this is the way the half of the given px option is.

02:01 And now in the first part of the question, it is given to me that p10, the profit made after the profit made after manufacturing some 10 blenders is a negative value, minus 263 .3.

02:19 So to break even, i need to make p of x equal to 0.

02:25 To break even, the profit has to be at least 0.

02:36 So in that case, tx equal to 0, so the first value of tx equal to 0, we get x equal to 25 .23.

02:46 So in the whole number term if we talk, i will say that the company has to manufacture at least 26 blunders.

02:58 So that's it.

03:00 To break even, the firm has to produce 26 blunders.

03:19 This is in whole number terms i'm talking...

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