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hero painted his room at a rate of eight square meters per hour after three hours of painting he had 28 square meters left to paint so after three hours of painting he had 28 square meters left to pain so they're talking in terms of how much we have left to paint not how much we have painted let a of T denote the area to paint a measured in square meters as a function of time T measured in hours so a of T once again this is how much we have to paint not how much we have painted right the function's formula so what I like to do is let's just think a couple about a couple point here let's just make it a little bit tangible for us so let me notice a different color so let's think about what a if T is at different times so this is time this is a as a function of time and they give us one of them they say after three hours of painting after three hours of painting he had 28 28 square metres left to paint and once again a of T is how much we have left to paint not how much we have painted so I'm going to leave some space here for some other valid may be 0 1 2 let's write 3 over here after 3 hours he had 28 square meters left to paint we're just assuming that this is going to be in square meters and this isn't ours now they tell us that he painted his room at a rate of 8 square meters per hour so let's actually back up a little bit let's back up let's say after after 2 hours how much would he have had left to paint how what would a ft bin would it have been more than 28 or would have been less than 28 well he's painting 8 square meters per hour so every hour that goes by he's painting more but a if T isn't how much he is painted it's how much he has left to paint so he would have left he would he should have less to paint as time goes up so as time goes up as that as time increases a of T a of T should go down so at 2 hours he should have more to paint than at 3 hours because remember is how much he has left to paint so how much more would he have had to paint at two hours than at three hours well it tells us that he has eight square meters - he paints at a rate of eight square meters per hour so between two and three hours he would have painted eight square meters so at two hours he would have had eight square meters more to paint so if you add 8 to this right over here you wouldn't be at 36 so you he would have had 36 square meters to paint at 2 hours and then what about 1 hour so in 1 hour he would have had 8 more Square Meters to paint so 36 Plus 8 that is 44 and at 0 hours what would he have had to paint so we just in another color at 0 hours well he would have had to paint 8 more square meters so 44 plus 8 is 52 let's think about whether that makes sense if right when he was starting he had 52 square meters to paint then an hour goes by so your change in time is 1 hour and then your change and how much he's had to have how much he has left to paint it goes down by 8 change in a is equal to negative 8 that makes sense his rate of change should be negative because the amount he has left to paint goes down as time goes forward so this was pretty interesting now let's see if we can actually construct a formula or can the formula that describes this function well this is happening at a constant rate every time T goes up by 1 we see a of T goes down by 8 T goes up by 1 a if T goes down by 8 they tell us that and that's because he paints it at a rate of 8 square meters per hour so whenever you're describing something that's happening at a constant rate that can be described by a linear function and a linear function will have the form a of T is equal to your rate of change times time plus wherever you started and M and B or just the letters that people tend to use for your rate of change your slope if you were graphing this and B where you start it off and this would be your vertical intercept is sometimes you call it your y-intercept but in this case would be your a intercept if we're talking if we're thinking about the actual it would help us find the intercept if we were graphing this thing but we actually already know both of these things we know what our rate of change is it is negative 8 I mean we could say well what's our slope our slope is change in change of a over change in T change in a over let me write it this way let me do it in a different color just for fun so our our let me our slope is just our change in our dependent variable over our change in our independent variable which is equal to negative 8 they tell us that it's equal to negative 8 so this thing is equal to negative 8 and B is going to be equal to a of 0 a of 0 well when T is equal to 0 this term right over here goes away and you're just left with B a of 0 is equal to B and we know what a of 0 is it is equal to 52 so we know this right over here is 52 and we're done we know then I'll just rewrite it just for fun a of T the area that they have that he has left to paint as a function of time is equal to negative 8 times time times time plus 52 plus 52 and you can confirm that the unit's make sense because this negative 8 this negative 8 and actually out let me write it one time with the unit's just because it is an important thing to think about area as a function of time this is how much he has left to paint is going to be equal to is going to be equal to negative 8 negative 8 square meters per hour so negative 8 meter squared per hour times T hours times T hours maybe I'll write out hours so you don't think it's a variable T hours let me write the hours over here T hours plus 52 square meters plus plus let me do it right over here plus I have trouble story switching colors plus plus 52 square meter and you see hours hours / hours cancels out you just be left with meters squared you'd have negative 80 square meters + 52 square meters and so the a of T is going to be given to you in square meters now there's other ways that you might have wanted to tackle this you might have immediately said hey look my rate of change is 8 square meters per hour now you have to be very careful there you have to say what you might be you might have said well my rate of change maybe is going to be positive 8 square meters per hour you have to be clear that a is not how much he's painting it's how much he has left to paint so his rate of change of how much heat he has left to paint is decreasing at 8 square meters per hour so you might have said okay immediately this my my formula would look like this a if T is going to be equal to negative 8 times T plus some B and then you could have used this information right over here to solve for B you say hey when T is equal to 3 when T is equal to 3 a is equal to 28 you could have just used this information right over here and substitute it right over here so when T is equal to 3 when this is 3 a of T is 28 and you have forgotten 28 is equal to negative 8 times 3 so negative 24 plus B and then you would have added 24 to both sides of whoops you would have added 24 to both sides and you would have gotten 28 plus 24 is 52 52 and then on the right side you would just have B you would get B is equal to B is equal to 52 which is exactly what we got over there I'd like to do it this way just to make sure that we really conceptualize we really grok what was going on