in this section we’re going to talk about the cost revenue and profit and if you recall in the previous unit we talked about the cost revenue profit using linear models. in this section we’re going to talk in particular about the revenue and profit using quadratic models now before all that let’s talk about something that we all know and some of us love – the movies. so a movie theater charges 13 dollars per ticket and sells an average of 934 tickets each night. as you’re doing this make sure you follow along with your media notes and fill them in please $13 it represents the price of the ticket go to movies buy a ticket, it’s $13… so $13 represents the price 934 so the movie theater is telling an average of so average you know we by that we need maybe on tuesday they’re selling fewer tickets friday and saturdays little more tickets but on average they sell 934 per night. so 934 represents the quantity of tickets that consumers are buying. now we’re going to think of quantity in another way. we’re going to think of quantity as the market demand ok so what is the market demand? in general market demand is a consumer’s desire and willingness to pay for a commodity ok so in this case consumers are willing to buy 934 tickets on average per night. right? so the demand for tickets is 934 so the demand is how many units of particular commodity are consumers willing to buy and in this case the man is 934 because that’s how many tickets on average consumers are purchasing on a nightly basis we are going to think of quantity as demand now the revenue of the movie theater so the revenue is how much money the theaters bringing in, right? so they’re charging thirteen dollars for each ticket and they are selling 934 tickets each night so the revenue is going to equal to $12,142. okay now remember revenue is not the same thing as profit. profit is revenue minus cost the 12,142 that just that just how much money’s coming in. alright now you know what happens every couple of years movie prices go up like when I was in college and high school tickets cost four dollars and then as time progressed you know to my horror they went up to $13 per ticket and now the prices are probably going to go up again. so we are going to look at that scenario in our new scenario the movie theater decides we are going to raise prices, and so when they raise prices the price goes from $13 per ticket to $15 per ticket. when I was in college and high school and tickets were 4 dollars per ticket, I used to go like twice a month. but when ticket prices started going to $13 I go like twice every six months to me it’s not worth it to go every single every single week or every single month even. so as price goes up the demand goes down the demand decreases so as the price increases demand usually decreases so fewer people are going to movies now that the movie tickets cost $15 so movie theater now sells an average of 821 tickets so the price of the ticket is fifteen dollars and the demand for tickets is 821 the revenue is price times quantity. so how much money is the movie theater bringing in? well they’re charging fifteen dollars . and they’re they’re selling 821 tickets, so the revenue is going to equal to $12315. this is great for the movie theater because their previous revenue with the thirteen dollars was 12142. by increasing the ticket price even though fewer people are going they’re making more revenue, which is $12315 ok now because price demand are related. as price goes up demand usually goes down as a price increases demand function. decreases we’re going to talk about something called the price demand function or we will refer to it, for now, as the price demand function but we’re going to refer to this out as price demand function of the price demand function it relates the price little commodity to his demand it relates the price to the demand so what happened as price increases usually demand decreases find first example it’s as the price demand function of producing ice cream is given by the following is given by p equals $OPERAND to 16 minus $OPERAND works where he is the price per unit of my screen and X is the demand for quality that consumers are purchasing in part a we want to calculate the revenue function for the skull honey so let’s think about this revenue you it is price times quantity so the revenue function revenue in terms of ice is given by the price we have to the price demand function which is one of six minus four X so the price is given by 10 6-4 eyes what’s the quantity not remember the price of a bunch of producing ice cream is given by P minus 2b was one of six minus four X where he represents the price X represents the units of ice cream sold for the man so this is the price than quantity of ice cream soul is going to be X units so we have the revenue is equal to the price times quantity and LSU vs so we’re gonna get the revenue equals to one of those days x minus or its okay its party and party you are given the cost function so the cost function of producing experience of ice cream is an X plus 3 using this you want to calculate the proper function now if you recall profit is the revenue how much money is coming in minus the cost the cost to produce certain of units of clarity ok so in this case are property of X is going to be the revenue which is one of success or square minus the cost which is tags Plus ok so we have profit equals to the revenue minus the cost let’s simplify this so the profit in terms of x equals 2 16 x minus one squared let’s distribute the negative about negative next three lists standard form minor like terms so profit in terms of the number of units soul of ice cream is given by negative 4x squared plus 96 x minus ok this is going to be a problem now what do you want to do it properly almost companies they want to maximize profit ok so now we want to figure out how many how many units of ice cream must be sold what lot of you must be sold for how many units in order to maximize the problem ok so in part c you want to maximize the problem in our previous lesson we learned that the maximum the minimum of a quadratic function occurs at the vertex how do we calculate the vertex the x-coordinate is given by x equals to negative V over 28 x coordinate of the vertex is given by x equals negative B over 2a so what are our AB&C just so it’s a reminder a coefficient of the x squared B is a coalition of nice and c is the constant so we’re going to look at the vertex of this quadratic function is going to be negative B which is negative 96 / 2 times 8 which is negative 4 so this is going to give you negative 96 / innovate that means this company must sell 12 units in order to maximize their their problem i’ll keep in mind you have to remember for what each variable represents X represents the back or the quantity of units that are sold so the helping yourself 12 units in order to maximize the problem ok now the next part of this question says what is the maximum profit well you know that excess 12 we can plug it to get profit so pm-12 to the profit when 12 units are sold that’s equal to negative 45 all square plus 90 days x 12 minus we probably despite a lot of simplifying this and get a value for that so ready to explore force multiply at the very end but the profit when you we sell 12 units is going to equal to five and seven dollars and that is going to be our maximum profit you sell less than this that the properties lower you sell more than this probably is also ok and a part d what should the company charge printed to maximize the profit so we’re looking for the price what price listed company charge order to maximize the profit so we give all this party we know the value of x is 12 when 12 units are sold the profit is maximized so what is the price when we solve X units all the price which fight always different than profit is going to equal to one of six minus four x 12 because selling penis finds profit so we’re going to p equals to 1 of 6 minus 48 and that will give you fifty dollars so if you price one unit of ice cream and fifty dollars so we’re looking at you know some big ice cream so what you do is you $58 then we will sell 12 units of that and our maximum profit will be five instead look at the next example example states the pricing and function of producing designer dog leashes is given by P equals 38 minus 2 it’s 4p represents the price index represents the units and publishes are sold a number of dollars for selling ok so party it says calculate the revenue now we know that revenue is price and quantity so the oppressive and function or the prices even by 38 minus 2x at this price what’s a quantity of God wishes that’s being sold well the price of a budget relates the price the quantity so this represents price ex represents quantity so the party is just going to be X is the revenue you distribute yes we’re going to get 3x minus 2-squared part b we want to calculate the profit function given the cost function of producing ex-emperor publishes is equal to 10 x + 7 so the profit function of producing X number of colleges given by the revenue which 30 x minus 2x squared minus the cost which is the next step ok so little more profit is going to be 30 days minus 2x squared let’s distribute the negative we have a negative next and you have a negative 74 the profit is standard form we have negative 2x squared and we got 38 x minus 10x which is going to be 20 X minus 7 ok so this is going to be a problem function with respect to the number of God wishes that all right part see how many units must be sold to maximize the profit so in Part C remember the maximum for the minimum value function the vertex the x value of the vertex is given by x equals to negative B over 2a so we have this case our values 28 so we have negative 28 are a value is negative 2 so we have 2 times negative 2 this gives us negative 28 or negative for you said so we have to sell seven minutes 47 leashes to maximize the problem now in part d what we have we have to figure out what is the problem so if the value nexus 7 that means we’re selling seven units then p of seven drivers as a prophet 176 old is going to be negative two x squared plus 28 times 2-7 ok so remember explore first x is practice again the Prophet way that these are soul is equal to $OPERAND ninety-nine dollars now this looks like a pretty low profit to be made but maybe probably just starting out maybe just someone whose lifestyle it’s you remember sometimes happens where when someone is for starting a business their profit is very low can on Part D what price to the company charge per unit in order to maximize profit well we said at seven units maximizes profit so what is the price when 70 this year sold prices going to be 38 minus 2x which is 57 that’s going to be a 38-14 so the price of each dog she’s going to be twenty-four dollars in prizes $24 publish is going to sell seven units which will maximize our profit at a new analysis