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A small bottling company finds that it costs $6,000$ to prepare 10,000 six packs of cola, and $8,000$ to prepare 15,000 six-packs. If each six-pack sells for $1.20,$ find and plot the graphs of the total cost function and the revenue function. Plot the profit function. Find the break-even point.

$$C(x)=0.4 x+2000, R(x)=1.2 x, P(x)=0.8 x-2000, x=2500$$

Algebra

Chapter 1

Functions and their Applications

Section 3

Applications of Linear Functions

Functions

Oregon State University

McMaster University

Harvey Mudd College

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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Determine the profit funct…

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working with applications of linear functions. We oftentimes want toe grab our cost, revenue and profit functions. Although we may only be given a couple coordinate points and a price to work with. That's all given at the top of our screen and written in black. I've formed a new X Y coordinate table just for ease of reference as we proceed. Let's start by graphing our cost function. And in order to do that, we're gonna want to know what our slope is. We can do that by using the slope formula being equal to y two minus y one all over X two minus x one. We can use our wives over here. We know we'll have 8000 minus 6000 on the top, which is 2000 written all over our exes of 15,000 minus six, minus 10,000, giving us 5000 on the bottom. All of which is equal to a slope of 0.4. That is RM value. Yeah. Next, we might wanna find in orderto complete our cost function. We want to find what our intercept is. We can do this by plugging some values into the cost function. We already have to see of exes, which are gonna be R Y values both 6000 and 8000. We just need to use one of them. So let's use 6000. We'll do 6000 plugging that in is equal to D, which is our intercept. And that's what we're searching for right now. It's the one variable that we don't have it the moment troops plus 0.4 which is RM value and are slope times are corresponding X value of 10,000. Working this out we get at 6000 is equal to indeed plus 4000 subtracting 4000 from both sides. We get that r D is equal Thio 2000. I'm with that. We have all the information we need to write our cost function which ends up being equal to our our intercept of 2000 plus our slope of 0.4 times X. Now, if we wanted to find exactly where these points on this line are here So you understand how I got the line toe look like that we know that we have an intercept of 2000, so that's got to be the point over on the y axis. And then because it's a linear function, all we need is one other point to be able to draw the line through. So we can just take one of the points that we have in our X Y coordinate table in the top, right? We can use 10,004 R X at the point where X is equal to 10,000. We have a Y value of 6000 Now. If the line weren't already drawn, you can see that you would be able to draw a point, draw a line through both of these points, and it would look very similar now if we wanted to find our revenue function. We know what our price is that was given to us as a dollar 20. So revenue function is equal to 1.20 x, which can be written down here. We know that that's what line where, what curve we're drawing. We just need to know how exactly to find it with revenue with the origin is always going to start at zero, because if you don't sell anything, you can't earn anything, and we just need one other point to draw it through so again we could take any one of our X values. Let's use 10,000 again. So our revenue, when X is equal to 10,000, gives us 1.2 times 10,000 ends up giving us the value of 12,000. So that gives us another coordinate point that we can draw this through. So when X is equal to 10,000, we'll have a Y of 12,000 so that we already have our X value on here. Just bring it up to meet the revenue line and wise 12,000. That is how you would graph your cost and revenue curves. And if we wanted to find out what our break even point Waas, that's simply the point at which these two curves intersect. It's all right in here, and we just need want to find what are X value is in order to find what that break even point is. So let's set our two curves equal to one another. Our 1.2 X, which is our revenue function setting that equal to our cost function being 2000 plus 0.4 x solving here we can subtract 0.4 x from both sides to get 0.8 x equal to 2000 2000, divided by 20000.8 is going to give us an X value equal to 2500 and that sits right down here. That is our break even point. And if we wanted to work with our profit function, profit is equal to revenue minus cost. We've already found a revenue and cost curso weaken dio our revenue function of 1.2 x subtract cost function of 2000 plus is your 0.4 x factoring out are negative. We end up with 1.2 x minus 2000 minus 0.4 x subtracting 0.4 x from 1.2 we get 0.8 x minus 2000 and that is equal to our profit function. Yeah, if we wanted to graph this, we can find what our we know that our intercept is from this equation here is going to be negative 2000 and all we need is one other point in order to grab this because again, it's a linear function so we can use any point that we'd like how, But we use our 10,000 once again plugging 10,000 and for our X value here ends up giving us your 0.8 times 10,000 minus 2000. Working that out, we end up finding that it is equal to 6000. So at the point where X is equal to 10,000, we have a Y value of 6000 on our profit function. And again, you would just draw a line to these two points and you would have drawn your graphic function. And that is how you graft both your cost function and your eminent functions on one graph, and you brought function on another as well as how to find your breaking point.

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