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Current time:0:00Total duration:9:00

Video transcript

Domonique from Dominique's pizza bakes the same amount of pizza every day she used to spend $8 each day on using the oven and a dollar 50 on ingredients for each pizza so eight dollars each day on using the oven and a dollar 50 on ingredients for each pizza one day the price for the ingredients increased from a dollar fifty to two dollars per Pizza Dominique made some calculations and found that she should make eight pizzas more each day so the expenses for a single pizza would remain the same and assume they're saying the total expenses for a single pizza because clearly the ingredients the ingredients cost is not the same we're talking with the total ingredients so if we were to spread the cost of the oven across all of the pizzas write an equation to find how or the total cost for the oven per day to spread that across all the pizzas write an equation to find out how many pizzas Dominique baked each day before the change in price use P to represent the number of pizzas so let's just think about her total cost per pizza before and then her total cost per pizza after if she bakes eight more pizzas so before we're going to use P to say this is that's the number of pizzas she baked per day before the change in price so before the change in price on a given day she would spend $8 on the oven and then a dollar fifty on ingredients for each pizza so one point five or dollar fifty times the number of pizzas this would be her total cost on all the pizzas in that day it's the oven cost plus it's the ingredients cost so if you wanted this on a per Pizza basis you would just divide by the number of pizzas now let's think about what happens after the change in price after the change in price for cost per day for the oven is still eight dollars but now she has to spend two dollars per pizza on ingredients so two dollars per pizza and instead of saying that she's making pizzas let's say that she's now baking eight more pizzas each day so it's going to be P plus P plus eight and so this is going to be her total cost for all of the pizzas she's now baking and so if you wanted on a per Pizza basis you well she's now making p+ 8 pizzas you would divide by P plus 8 and the problem tells us that these two things are equivalent here you had a higher cost of ingredients per pizzas but since you're in since you are now baking more pizzas you're spreading the oven cost among amongst more and more pizzas so let's think about what P has to be if P is has to be some number some number of pizzas so that these two expressions are equal her total cost per pizza before which he only made P is going to be the same as her total cost per pizza when she's making P plus 8 pizzas so these 2 things need to be equal so we did that first part or we did what they asked us we wrote an equation to find out how many pizzas Dominique baked each day before the change in price and we used P to represent the number of pizzas but now for fun let's actually just solve for P so let's just simplify things a little bit and actually so this part right over here actually let's just cross multiply this on both sides or another way of thinking is multiply both sides times P plus 8 and multiply both sides times P so if we multiply by P plus 8 and we multiply by P we multiply by P plus 8 and we multiply by P that cancels with that that cancels with that on the left-hand side so let's see we have to just do the distributive property twice right over here what is P times 8 plus 1.5 P well that's going to be 8p I'm just multiplying the P times this stuff first plus 1.5 P squared and now let's multiply now let's multiply the 8 times both of these terms so plus 64 plus 8 times 1.5 that is 12 plus 12 P and that's going to be equal to let's see let's multiply P times all of this business so that's going to be equal to 8 p 8 times P is 8 P and let's see I could distribute these terms and then multiplied by P so 2 times P is 2 P times P is 2 P squared + 2 P squared and then 2 times 8 is 16 times P is 16 P so now we have well we essentially end up with a quadratic equation but let's simplify it a little bit so that we act it or apply the quadratic formula so let's see let's let's throw let's let's subtract - P squared from actually let's subtract 1.5 P squared from both sides so subtract 1.5 P squared subtract 1.5 P squared actually let me just put everything on the left-hand side just because that's might be a little bit more intuitive so let's subtract 2 P squared from both sides subtract 2 p squared from both sides let's subtract let's subtract 16 P from both sides so we have an 8p + 12 P and then we're going to subtract a 16 P from both sides and then or actually let's subtract an 8 P as well from both sides we have the 16 P and an 8 P so that actually works out quite well so now we've subtracted 8 P from both sides 16 P from both sides we've still essentially subtracted all of this stuff from both sides and we are left with let's see 1.5 I'll start I'll do in degree order 1.5 P squared minus 2 P squared is negative negative 0.5 P squared now let's see these cancel out 12 P minus 16 P is minus 4 P and then we have plus 64 plus 64 and then that is going to be equal to 0 that is going to be equal to 0 and just to simplify this a little bit or just make this a little bit cleaner let's multiply both sides of this equation by negative 2 by negative 2 I want the coefficient over here to be 1 so then we get P squared + 8 P P squared + 8 P is going to be equal to is going to be equal to let's see negative negative times negative 2 so minus 128 is going to be equal to 0 so let's see if we can factor this can we think of two numbers where if we take their product we get negative 128 and if we were to and if we were to add them together we get positive 8 so they're going to have different signs right over here so let's see if we say 12 times let's see 12 times well let's see what can we what numbers could this be so if we were to think about 128 is the same thing as let's see 16 let's see 16 goes into 128 we work through this so 16 goes into 128 does it go 8 times 8 times 6 is 8 times 6 is 48 8 times 10 is 80 plus 40 is 128 yep goes 8 times so 16 and 8 seem to work so if you have positive 16 and negative 8 they would product would be negative 128 so we can factor this out as P plus 16 times P minus 8 is equal to 0 now this is going to be equal to 0 if one of these two at least one of these is going to be equal to 0 so we have two solutions either p plus 16 is going to be equal to 0 or P minus 8 is equal to 0 this one right over here subtract 16 from both sides you get P is equal to negative 16 here you get P is equal to 8 if you add 8 to both sides now we're talking about a number of pizzas made so we're not this one doesn't apply this would be like her Dominique eating 16 pizzas or somehow destroying 16 pizzas today we're not interested in that solution so if we want the solution to the original question the number of pizzas she made before the increase in price she made 8 pizzas she made 8 pizzas per day so P right over here needs to be equal to 8 so before the change in price she made 8 pizza day after the change in price she made eight more pizzas a day or sixteen pizzas per day