Market Research A market analyst working for a...

Market Research A market analyst working for a smallappliance 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 smallappliance 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

Graphical Analysis in Business

Graphing the profit function provides a visual representation of how profit changes with different production levels. It aids in identifying key features such as where the function crosses the horizontal axis (i.e., the break-even point), intervals of loss and profit, and turning points. Using an appropriate viewing rectangle ensures that the significant behaviors of the function, like its maximum or minimum values, are clearly observable.

Optimization in Profit Analysis

Optimization in the context of profit analysis is focused on finding the production level that yields the maximum profit. By analyzing the graph of the profit function or utilizing calculus techniques, one can identify the local maximum point where profit is at its highest. This optimal point informs decision-makers about the most advantageous production level under the modeled conditions.

Profit Function Modeling

A profit function models the relationship between the profit of a business and the quantity of goods produced or sold. In many real-world problems, this function is a polynomial that combines linear, quadratic, and cubic terms to capture various effects such as fixed costs, variable costs, and diminishing returns or other nonlinear phenomena. Understanding the structure of the profit function is essential in analyzing business performance across different production levels.

Break-Even Analysis

Breaking even occurs when the profit is exactly zero, meaning the total revenue equals the total costs with no net gain or loss. Calculating the break-even point involves setting the profit function equal to zero and solving for the production quantity. This analysis helps determine the minimum level of production needed to avoid a loss, which is crucial for decision-making in business operations.

Transcript

00:01 So we're looking at this profit equation, and they're selling blenders.

00:04 So we have p of x, so the total profit they're going to make is 8x plus 0 .3x squared minus, to move my paper out of away, 0 .0013x squared and then minus 372.

00:23 And that's going to be in dollars.

00:26 And we have to graph it.

00:28 So i put this in as y sub 1 in my calculator.

00:31 And then i did a table of values.

00:33 I made a table of values.

00:35 I started at zero, and because i don't know how high to make this graph go and how low.

00:40 I can see it's going to at time zero, they're going to be losing $372, but how low do i go, how high do i go? and so i scanned through, and again, i had my calculator start at zero for x, and then i had it go up by tens.

00:56 And i could have gone up by something larger than that, but i didn't have any idea how many blenders this place would sell.

01:04 And it ends up, i could see that the graph, i know it's a, oops, and this is supposed to be a three.

01:10 I know it's a third degree polynomial with this coefficient being negative.

01:15 So i actually know that the graph is going to start high.

01:18 It's going to do something like this and do that.

01:21 Now, how much of that graph are we going to see? you know, i don't know.

01:25 But i found that my best window was to start.

01:30 At zero and i went up to and let's see i went up to 300 and i had it put tick marks or put the x scale every 100 and then i went as low as negative 500 i didn't have to go that low i could have stopped at like negative 372 and i went up to 4 ,000 because i could see that it looked like it peaked at 3 ,000 something and on my calculator i have it grabbed and it kind of looks like this and it comes up and i can see it goes down.

02:04 All right.

02:05 And so we're seeing, you know, part of the graph like in here.

02:09 And so we had to find where was the break -even point.

02:13 So we know that here is our first break -even point, that you will make $0.

02:19 So we want to find that.

02:20 And i'm going to find that on my calculator.

02:22 I'm going to do that by, i could zoom in on this spot, but i'm going to hit on my calculator second in calculate.

02:29 And i want to hit a zero.

02:32 And then i have to make it left bound...