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problem five at the beginning of 2010 a landfilled contained 1,400 tons of solid waste the increasing function W so I guess W constantly increases the increasing function W models the total amount of solid waste stored at the landfill planners estimate that W will satisfy the differential equation so the derivative of W with respect to time is equal to 1 over 25 times the quantity W minus 300 for the next 20 years W is measured in tons T is measured in years from the start of 2010 alright let's get into this so Part A use the tangent line to the graph of W at T equals 0 to approximate the amount of solid waste that the landfill contains at the end of the first three months of 2010 so since time is in years that would be three months would be 1/4 of a year so T equals 1/4 so it seems at first is really daunting they gave us a differential equation we don't even know what W what the actual function W is how do we figure out its tangent line but we just have to kind of think about what they're asking so regardless of what W looks like so let's think about it a little bit so we don't know yet what W actual eat what W actually does look like but they tell us it's an increasing function so at time 0 it has 1,400 tons in it they tell us that right up here and that it increases we don't know what the function actually does look like but let's say that that is W so what they're saying in problem a is they're saying use a tangent line to the graph at T equals 0 so there's let me actually many drunk I'll be a little bit differently so W might look like this W might look like this so they're saying it's fine the tangent line find the slope of that tangent line at T equals 0 so there's some slope right over there and then we can use to approximate we can use this slope to kind of create a linear approximation of where we're going to be a quarter of a year from then so if you go so although we don't know what W is just yet we could take the slope of this line out to T is equal to so this is our T axis to t is equal to 1/4 and wherever wherever that line takes us out after a fourth of the year that will be at least a decent approximation we're extrapolating forward from that first point and and the current slope so we really just have to figure out the slope of this line and just see where that line is at t is equal to 1/4 and you say wait how do we know how do we know what the derivative of W is at 0 without knowing W and that's where we can go straight to this differential equation we could actually rewrite this differential equation using slightly different notation this is the derivative of W with respect to T we can write that as W prime of T is equal to 1 over 25 1 over 25 times the function W which is a function of t minus 300 and when you look at it this way it becomes a little bit clearer on how to figure out what the derivative of W is at 0 we literally the slope of this line the slope of that line literally is just the derivative of W evaluated at 0 so let's literally take the derivative of W and evaluate it at 0 so we have W prime of 0 is equal to 1 over 25 times W of 0 which we know we know this is 1,400 tons of solid waste this is how much waste there is at time 0 - minus 300 so this right over here is 1400 and so we have W prime at time equals 0 so our slope at time equals 0 our derivative at time equals 0 is equal to 1 over 25 times 1,400 minus 300 is 1,100 and 25 goes into 1,100 it goes into 1144 it would go into it forty four times right it goes four times into each hundred we have 1,100 sear so it'll go forty four times so it goes forty four so the slope here the slope of this line the slope of that line is forty four we could say M for slope or actually let me just write down the word the slope of this line is forty four and I just ate some peanuts or something so my voice is a little dry so bear with me but the slope of this line is twenty four so how do we use that to find an approximation for the amount of waste at the landfill at the end of the first three months well let me zoom in a little bit and actually my fingers are all salty too so maybe I don't have a proper grip on my pen but I'll try my best so let's say that let me just zoom in a little bit more so we're starting at 1,400 tons so this is my W axis this is my time axis we're starting at 1400 tons and we are increasing from there they're telling us that it is an increasing function so maybe it looks maybe it looks something like that and that our slope our slope right over here and I don't know I haven't drawn W exactly that that accurately I'm just guessing what it might look like at this point the tangent line has a slope of 44 so that is our tangent line and what that's saying is if we go out 1 if we go out one unit of time which is one year then we would have gone up 44 in tonnage so if we if we if we use this line as approximation after one year this point would have fourteen hundred and forty four tons but we're not trying to approximate a year out we're trying to approximate 1/4 of a year out so we're trying to approximate so this would be half a year out this is 1/4 of year out we're trying to approximate we're trying to approximate that point right over there so it's going to be 1400 this point right over here and we could write down the equation of this line if we like we could say this line this line so we'll call this you know the W approximation because this isn't exactly our W function this is equal to the slope of our line 44 times time plus our W intercept we could say our plus our initial condition plus 1400 so if you put time is equal to 1/4 in there you get is equal to 44 so let me say we write it this way so our approximate W at time is equal to 1/4 of a year is equal to 44 times 1/4 plus 1,400 running out of space and my salty hands are having trouble writing this properly and so 44 times 1/4 or 44 divided by 4 is 11 plus 1400 and so you add them together you get 14 hundred and eleven or 1400 and eleven tonnes this is our approximation we just took the slope from our starting point and use that slope as an approximation it probably is not the exact amount of tonnage based on the actual function W but it's a it's an OK approximation but that's our first answer for Part A 1400 and 11 tons

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