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# Worked example: accumulation of change

AP.CALC:
CHA‑4 (EU)
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CHA‑4.A (LO)
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CHA‑4.A.1 (EK)
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CHA‑4.A.2 (EK)
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CHA‑4.A.3 (EK)
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CHA‑4.A.4 (EK)

## Video transcript

it took 20 minutes before Jughead noticed that his hot tub had sprung a leak once he realized it he opened the hot tubs drain and the rest of the water rushed out in 40 minutes the rate at which the water drained from the hot tub in gallons per minute is shown how many gallons of water were in the hot tub before it started to leak so let's see what they have over here so they've plotted time or they probably gallons per minute versus time within minutes and so we see here this blue line this is the rate at which water drained from the tub rate which water drained from the hot tub so at minute zero the water wasn't really draining from the hot tub and then more not just more drained but the rate at which the water was trading increased so ten minutes into his bath it was the the water was draining at a gallon per minute and the 20 minutes into his bath water was great draining at two gallons per minute and then he noticed it and he opens the drain I guess he wants to accelerate the end of his bath so he opens the drain and then all of a sudden water starts draining out at a much higher rate at 20 gallons per minute but then that decreases and so we could think about physically why why that might decrease maybe there was just a it was just less pressure or whatever we're not gonna go into the physics of it but we're just going to take this chart as fact but the rate at which the water drains decreases all the way to the 60th minute which is 40 minutes after he opened the drain 60th minute all of the water was drained all of the water was actually drained out so given that how do we think about how many gallons of water were in the hot tub before it started to leak and I encourage you to pause the video and try to see if you could figure it out yourself before I work through it well to answer this question how many gallons of water in the hot tub before it started to leak that's this answering the same question as how much total hot water drained out so how much how much drained drained out and to think about that we can just kind of go back to what we knew before we learned about calculus if I have something happening at a fixed rate so let's say that this is let's say this is gallons per minute so this is still the same the same the same context I guess you say this is time in minutes now let's say things are draining out at a constant at a constant rate if you wanted to figure out how much drains out over a certain interval of time let's say this interval of time right over here let's call that delta T over delta T you would just multiply you would just multiply the rate over that interval of time the rate over that interval of time which would be the which would be which we could represent by this orange height right over here times the amount of time that passed by which essentially would give you the which would give you the area under the curve over that interval the area under the curve over that interval this this area would tell you the gallons that drained over that delta T and this doesn't just apply when you have a constant when you have a constant rate if your rate look something like this as we've seen in other videos as we've seen in other videos you could figure out the amount the amount that has drained the amount that has drained in a specific interval let's say this interval right over here by x essentially figuring out the area under the curve over that over that interval and the units work out if you multiply gallons per minute times minute your this area is going to be in terms of in terms of gallons this area is going to be in terms in terms of gallons another way you could think about it and this goes back to well how do you figure out the area of a trapezoid right over here well the area of a trapezoid you could find the average height of the trapezoid which would be the average of the the beginning and the end period you take that average over there and this is this would work for a line like this if you take that average height and multiply by by the change in time you are going to actually you're going to figure out you are going to figure out that area and that's another way of thinking another way of thinking about it is you're taking the average rate over that interval is the interval and that's going to give you the total number of gallons and so we just have to apply that idea over here we just literally figure out the area under the curve over the entire interval when the water was actually leaking or draining so essentially the area under the curve between zero minutes and 60 minutes and so it's going to be this area all right so that in magenta plus all of this area under under this part as well and to help us think about that I'm just going to split that up into some sections so I have this triangular section up here so I have this triangular section I could just think about this as a trapezoid but I'm just gonna split it up into a triangle and a rectangle and then I have and then I have this section right over here in green so what is the area of this entire thing well the area right over here we have 20 gallons per minute so 20 or sorry not 20 gallons per hour 20 minutes 20 minutes times 2 gallons per minute 2 gallons per minute times 1/2 times 1/2 gives us the area under this triangle so that's going to be that is going to be 20 20 gallons we see that the units work out nicely and so that's essentially how much this drained out in the first 20 minutes and then this green area is going to be 40 minutes 40 minutes times 10 gallons per minute times 10 and actually maybe you know the unit's since I'm breaking it up in this strange way I'll just multiply I'll just figure out this this area in a unitless way so 40 times 10 which is equal to 400 and then finally in blue I have 40 times this height right over here between 10 and 20 is another 10 but I'm going to multiply that times 1/2 so it's going to be 40 times 10 times 1/2 which is going to be 200 and so you add all of these together these areas together you're going to get 400 plus 200 is 600 plus 20 you're going to have six hundred and twenty gallons 620 gallons gallons is how much water and total drained out or how much water was in the tub before it started to leak or the hot tub that this is quite big for a regular tub but this could be like immunological you're a hot tub or something