 So in the previous video, we introduced the idea of linear modeling using linear functions to model growth or decay of data I want to do some more examples of that in slightly different contexts these other examples will go through a little bit quicker than the first example because we were kind of explaining everything and Honestly speaking the most important part of a story problem is setting it up correctly If you feel like if you can set up a story problem correctly Most of the algae will probably be fairly straightforward in this example We have a particular city dump which has a maximum capacity of 20,000 tons of garbage It currently is holding 8,000 tons of garbage and each month the garbage collectors bring in 120 tons of garbage into the dump So if this current growth rate, which is constant remains Constant at which point will our city dump reach its capacity, right? So we have to kind of know this so that we know we have to build a new dump or expand the current dump Right? We have sort of a time limit before you know street garbage just starts piling up in the street here So we want to come up with a linear function, right? So our if we're adding a hundred and twenty pounds To 120 tons of garbage per month, right? This is a constant growth This is a constant growth and when you have constant growth that means this is in fact linear growth We can model our data using linear functions So our function is going to look something like f of x equals mx plus b and as a little bit of a side I typically will set x equals zero to be the current the current time So where we are right now, so that means we are currently at 8,000 And so that's actually going to give us our y-intercept. Our y-intercept is currently 8,000 when x equals zero We're at 8,000 x will be a measurement of time And so we talk about months, right? And so we should be measuring x here We should be measuring this is some time measurement in months X will measure how many months have elapsed from the current moment All right, so now we have the y-intercept We actually have to find the slope of this thing and the slope is actually given to us, right? We're adding a constant rate of 120 tons of garbage per month And so that gives us our slope value m equals 120 tons per month Ideally the unit used to measure the x should be the denominator of your slope Which is a rate tons per month that way if you take tons per month and you times it by months That will give you tons of garbage You know literally and figuratively there, right? And so then our function would look like f of x is going to equal a hundred and twenty x plus 8,000 This gives us the model. This is the model. This is the toy we can now use to make predictions It the capacity is 20,000 so we then set this equal to 20,000 f of x is equal to 20,000 We want to solve for x so we get 120 x plus 8,000 And then we proceed to solve this equation Subtract 8,000 from both sides. We see that 120 x is going to equal 12,000 and Then divide both sides by 120. We're going to get that x equals 100 x equals 100 here so this means that in a hundred months In a hundred months the Group that we expect that are dumped to be reaching capacity are very close to it kids again This is just an estimate, right? We can't expect perfect 120 tons each month, but this is an estimate, right? So then we would answer this in something like the following so we we predict we predict that in approximately Approximately eight years four months Which be aware that's that's that's a hundred months, right? If you divide a hundred by twelve you're going to get eight remainder four so in Approximately eight years in four months the the town dump Will reach full capacity Which is sort of bad news if we don't want Trash is spewing all over the city, right? So basically our town has eight years to come up and implement a plan to fix the Garbage situation on whatever that whatever that is And so we can use linear growth to model this type and model model and solve this story problem here And the critical thing here is that we use the linear model because constant growth is linear growth The two things are one and the same thing