Market Research A market analyst working for a small-...

Market Research 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 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?

Market Research 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 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

Graphing Polynomial Functions

Graphing polynomial functions involves plotting the function on a coordinate system to visually analyze its behavior over a range of values. This method is useful for identifying key features such as zeros (break-even points), local maxima, and local minima, which are crucial for understanding trends in profit or cost structures as production levels change.

Cubic Profit Functions

Cubic profit functions are mathematical models represented by third-degree polynomials that are used to describe the relationship between the number of units produced and the total profit of a company. Such functions typically show regions of increasing and decreasing returns as production scales, and they are useful for capturing the nonlinear dynamics often present in real-world economic scenarios.

Break-even Analysis

Break-even analysis involves finding the production level at which total revenues equal total costs, resulting in zero profit. In the context of a profit function, this analysis focuses on determining the values of the variable where the function equals zero. This helps businesses understand the minimum level of output required to avoid losses.

Optimization in Business Context

Optimization in a business context refers to the process of finding the production level that maximizes profit or minimizes costs. This often involves identifying critical points on a profit function, either through graphical methods or calculus-based techniques, to determine where the function attains its maximum or minimum values.

Transcript

00:01 So we've got our equation here, p of x equals 8x plus nt1 .3x squared minus 0 .0 .03x to the power of 3 minus 3172.

00:10 X equals the amount of blenders that are sold and p equals the profit.

00:15 So part a here is asking, they make a loss when just a few blenders are sold.

00:22 So it's asking how many blenders must have been produced to break even.

00:26 Okay.

00:27 So we're essentially need to find here.

00:31 At what point does the profit start to become greater than zero? and so it's going to be minus.

00:47 We'll put this into, oh, so essentially we need to find our, okay, we're going to draw, we're going to start off by drawing a graph, i think.

00:59 Yeah, so we're going to start off by drawing a graph.

01:03 I think i can see where this is heading, yeah.

01:06 So we've got a negative cubic graph here.

01:10 So we're going to draw this.

01:13 So it's going to come from above.

01:16 We're going to get one turning point roughly here maybe and then another turning point up here.

01:21 And we need to find this point here.

01:26 I imagine though the, sorry, i imagine this green line here would probably be more over here in this example because you can't really build negative things.

01:37 So we need to find, sorry, it'll probably be round about there when it starts to dip.

01:45 We need to find this point here.

01:48 What is this point? so we're going to find our second x intercept.

01:56 So when x equals zero.

01:59 So we're going to set the equation equal to zero.

02:02 I'm going to solve it.

02:05 So p of x equals.

02:07 And we can see on the graph, that it's actually going to be the second time.

02:12 That's why the graph is so important here.

02:14 We're not going to take our first, and we're not going to take our smallest value.

02:18 We're going to take our second largest value.

02:21 And that is going to be our answer.