[Music] them two quadratics I like quadratics because they lead to factoring so let's get going a quadratic in one variable is an equation of second degree that can be written in general form as ax squared plus BX plus C equals equal to zero where a is not zero what happens if a is zero then what kind of equation do we have a line of course a b and c all represent constants how do we solve ax squared plus BX plus C equals to zero well we have three methods one is my favorite called factoring 2 is called extracting roots and three is the quadratic formula so we've already talked about factoring we have a whole video over factoring so if you need to go back and look at that please do in this video I'm going to be doing extracting roots and quadratic formula because I don't think I have used either one of those methods yet and I want you to be able to see them now to graph we're going to use the vertex the vertex is either the maximum point or the minimum depending on whether the graph which is a u opens up or goes down the axis of symmetry is the line that goes through the vertex and it's a vertical line because we're only going to be talking about quadratics that go up or down if the function ax ax squared plus BX plus C then has a Vertex we find it by doing x equals negative B over 2A and then plugging that x value back into the function so we have f of negative B over 2A to find the Y value and that is the vertex if we have y equals ax squared plus BX plus C then if a is positive it opens up if a is negative then it opens down so that's all the information that we have or that you need to know let me say it that way that you need to know about what about parabolas there's a lot more out there that we use in other classes all right the amount of Airborne population p from a power plant depends on wind speed s among other things with the relationship between P and S approximately approximately by P equals 81 minus .01 s squared find the values of s that will make p equal to zero what value of s from part A makes sense in the context of this application what does p equal to 0 mean in this application find the values that will make p equal to 0 so that means we're going to set this equation equal to zero and solve so I'm going to use extracting Roots which is not something I've done so far so that means we move now it looks like I moved the 81 to the other side it doesn't matter what you want to do so I could have just as easily what I really did is I moved this to this side to the other side the 0.01 s squared to the other side but remember that if I have 2 equals X plus one I can also write that as X Plus 1 equals 2. it doesn't matter the order in which I write things okay so just don't freak out because I wrote 81 equals 0.01 s squared instead of 0.01 s squared equals 81. okay don't let that bother you so the first thing we're going to do is divide by 0.01 and I get 81 100 equals that s squared now I'm going to take the square root of both sides when you take the square root of both sides which is what extracting roots does you put that plus or minus and the only reason I don't do extracting Roots a lot of times is because students forget to put that plus or minus so you put that plus or minus now I'm going to ask a question a lot of students are like okay I gotta have my calculator now I gotta have my calculator now oh my gosh calculator okay no you don't what is the square root of 81 9. okay so where's my so if I'm looking at 81 this and I just want to take the square root of that okay so I know the square root of 81 is 9. now what is the square root of a hundred ten then this whole thing right here is 90. it's that simple okay I am a number Theory person all the way so I can factor I can do number Theory I very seldom ever use a calculator so just telling y'all that's the easy way to do square roots all right so the next one says this is plus or minus 90. I don't want y'all to think where did she come up with it that's how square root of 81 is 9 square root of 100 is 90. this 8100 this is 81 times a hundred so you've got 9 square root of 100 okay so that gets us the first part the next part says what value of s makes sense in this context for this for this application well s is the amount of pollution now we all walk outside we all live in the world or at least I hope we do I am going to say this that it doesn't make sense to have negative pollution so then only positive values make sense so that means that s equals 90. so I'm going to click back a few slides and then I'm going to come back forward a few slides because I know where we're at the amount of Airborne pollution from a power plant depends on the wind speed among other things with the relationship of P to S being proximated by this so this whole thing is p is the pollution and S is the wind speed so if I put 0 in here I'm going to get 81. okay and if p is zero so the if if the amount of pollution is zero then what I'm looking for is the wind speed okay so the wind speed is 90 when the pollution is zero so when p is zero that doesn't make sense the particle level is at zero oh it's asking what it means it means the particle level is zero or the pollution this should be pollution that should be an end if I could write um this means the pollution the pollution particles are zero so but it's for when this study started so it's not that that there's no way that the pollution in the world was Zero at that point that just means that the level that they recorded that day is zero so in other words when we get to the regression which is the next lesson we don't start measuring time you may say the time of my birth is 19 so I was born in 1971. I'll admit that hey I was born in 1927 so I might take a thing and say your zero is then is 1971. so that is my year zero for my life or I might say that 1900 is year zero because that's when we started counting the time or started measuring time for a particular thing a particular thing we're studying or we might start measuring time in year 2000 and that might be when we started measuring time for this or and in 2019 might be when they started measuring the covid a virus and so that time time zero for that particular study is done at that so what this means P equals 0 is that that pollution level is P0 at that time so that's what they're talking about is that's when that pollution level was at zero not necessarily all pollution was gone at that point so I just want to make sure that y'all understand that it doesn't mean that all pollution was Zero it just means that they're taking the pollution levels at that time to be zero to be their starting point for measuring pollution if the demand function for a product is given by P equals 175 minus 0.2 X and the revenue function is given by R equals P of X where X is the number of units sold and P is the price per unit what price will maximize the revenue so in other words the price function is 175 minus 0.2 X so first we're going to want to find the revenue so the revenue is going to be 175 minus 0.2 x times x which equals 175x minus 0.2 x squared which is minus 0.2 x squared plus 175. this tells us that a is negative 0.2 and B is 175. how do we maximize the max is x equals negative B over 2A which is negative 175 over 2 times negative 0.2 which putting that in our calculator gives us 437.5 yes notice what the question is asking what price will maximize the revenue it doesn't ask what is the maximum Revenue in all the years that I've been teaching this class everybody says I put my answer into the revenue function it tells me I'm wrong I put my answer into the revenue function tells me wrong did you read the question not trying to be snarky but I want to make sure that you find your mistake it's important you will remember your mistake if you find it if I tell you where the answer is then you're going to keep making the same one um what price not what is the maximum Revenue but what price so what you're trying to find is the price so you're not putting it back into the revenue you're putting it back into the price function I want to know how much to sell my product for in order to get a maximum Revenue and it's 87.50 so P equals 175 minus 0.2 times 437.5 gives me 87.50 okay now ah there's too many words on the page there's not suppose a company has fixed cost of fifty thousand four hundred dollars and variable cost per unit of 4 9 X Plus 222 dollars where X is the total number of units produced suppose further that the selling price of its product is two thousand fifty dollars minus 5 9 x dollars per unit a find the break-even points B find the maximum Revenue C form the profit function from the cost and revenue functions D find the maximum profit e what price will maximize the profit huh Okay so what I have done is I have taken their A B C D and I have put them in a logical order so first you need to find the cost function then you need to find the revenue function then you need to find the profit function then you need to use the profit function to find the break-even points then you need to find the maximum Revenue then you need to find the maximum profit then you need to find the price and we're going to do this slowly suppose a company has fixed cost of 50 thousand four hundred dollars and variable cost of 4 9 X plus two hundred and twenty two dollars where X is the total number of units produced suppose further that the selling price is two thousand fifty dollars minus five ninths X dollars per unit form the cost function cost is variable cost plus fixed cost remember variable cost is cost per unit times the number of units so that means you take the 4 9 X Plus 222 times X plus 50 400. because that variable cost is per unit this right here tells you that this is V of x so then we get 4 9 x squared plus 222 x plus 50 400. form the revenue function revenue is price per items times the number of units our price per item so our little p of X which we will use at the very very end again is 50 20 50 minus five nines x times x equals 50 20x minus 5 9 x squared I'm going to say this remember back at the beginning remember here where I said that you're going to set your form the prophet and use the profit function this should tell you that your profit is a quadratic other than the fact that we're in the quadratic section okay so now let's go on so form the profit function so profit is revenue Minus cost so our Revenue function minus our cost function so that's 20 50x minus 5 9 x squared minus 4 9 minus parentheses 4 9 x squared plus two hundred and twenty two dollars X plus 50 40 50 400 close parentheses Distributing we get 2050x minus 5 9 x squared minus 4 9 x squared minus 222x minus 50 400. combining a like terms our final quadratic for profit is minus x squared plus 1828 x minus 50 400. next we want to find the break-even points and we want to use the profit function because it is a quadratic so we have a equals negative one b equals 1828 and C equals negative fifty thousand four hundred I'm going to go straight to the quadratic formula so x equals negative B plus or minus the square root of B squared all over 4AC the square root of B squared minus 4AC all over 2A putting in my numbers I have negative 1828 plus or minus the square root of 1828 squared minus 4 times negative 1 minus 50 400. all of that is over 2 times negative one simplifying it down I find that I get 28 and 1800. the company will break even when they sell both 28 units and 1800 units okay I do do the maximum Revenue so the maximum Revenue I have a equals negative 5 9 and b equals 2050. so I have x equals negative B over 2A so that's negative 2050 over 2 times negative 5 9. which equals 1845. the revenue evaluated at 1845 is 1 comma eight nine one one two five find the maximum profit uh profit is negative x squared plus eighteen twenty eight X minus 50 400. x equals negative one b equals eighteen twenty five x equals negative B over 2A which equals negative 18 28 all over 2 times negative 1 which simplifies down to 914. profit evaluated at 914 equals 7 8 4 comma nine nine six so when they sell 914 units they will maximize their profit to be seven thousand 784 996 dollars what price will make a maximum profit that means you take the 914 and put it into the price function that you were given in the statement of the question that function is 2050 minus 5 9 x so you get P of 914 Little P of 914 is 2050 minus 9 5 9 times 9 14 gives you 1 542.22 when they sell 914 units their price needs to be 1542.22 to maximize their profit thank you let me know if you have any questions [Music]