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Video transcript

a lake near the Arctic Circle is covered by a 2 metre thick sheet of ice during the cold winter months when spring arrives the warm air gradually melts the ice causing its thickness to decrease at a constant rate it's gonna decrease at a constant rate after three weeks the sheet is only one point two five metres thick after three weeks the sheet is only one point two five metres thick let s of T denote the ice sheets thickness s measured in metres as a function of time measured in weeks write the functions formula all right so we've got a we have some interesting things here they've given us some values for this function we know when time is equal to zero we know that s of zero when time equals zero that's when the sheet is two meters thick so the s of zero is equal to two and they also tell us that after three weeks the sheet is only one point two five meters thick and when we sit when we have the function s of T s is measured in meters time is measured in weeks so after zero weeks where two meters thick and then they tell us after three weeks so sf3 after three weeks we're one point two five one point two five meters thick or another way to think about it another way to think about it I could write T here in weeks and s in meters and when time is zero or two meters thick and when time is one point a second when time is three weeks when time is three weeks we are one point two five meters thick so when our change in time our change in time is equal to positive three we increased our time by three what's our change what's our change in thickness our change in thickness the triangle here this is a Greek letter Delta the shorthand for change in well this was negative 0.75 negative 0.75 so what was the rate of change over this time and they tell us that the rate of change is a constant rate so whatever it is between these two periods of time between zero weeks and three weeks it would be that same rate it would be that same rate between any two periods of time between zero week in one week or one week in two weeks or one and a half weeks and one point six weeks so what is the rate of change what is the rate of change of thickness relative to time well it's going to be change in thickness over change in time how much does our thickness change per time what we saw right over here our thickness went down set 0.75 meters 0.75 meters in three weeks in three in three weeks or we could say that this right over here is equal to let's see seventy-five divided by three is 25 so 0.75 divided by three is 0.25 and we have the negative out there negative 0.25 meters per per week so how can we take the information we have and express this as a function it's going to be a linear function because we see that we are changing at a constant rate let's think about it a little bit linear functions one way we can write it is in so we could write it if we were dealing with X and y you might recognize Y is equal to MX plus B often write written a slope-intercept form this is when you're dealing with X X's Z I guess you could say the independent variable Y is the dependent variable and B would be where you start what happens when x equals 0 and M is your rate of change it's your slope so in this case we're going to have we're not we don't have Y and X we're going to have s and T we have s as a function of time and it's going to be equal to the rate of change times time plus where we started plus B now what is B going to be what is B going to be well one way to think about it is well what's a what's a subzero going to be of 0s of zero is going to be M times zero plus B s of zero is going to be B well we already know that this ice sheet it starts off at two meters thick so s of zero is equal to B is equal to two so B is equal to 2 and what is M well we've already said that's our rate of change that is our slope that is how much are our depth how much our thickness changes with respect to time and we already figured out that that's negative 0.25 so M is negative 0.25 you could say that M is the slope between this point between the point 0 comma 2 and the point 3 comma 1.25 if we were plotting these points on a TS coordinate plane so now we can write what the function is going to be and maybe I'll do this in a new color just for fun s of T thickness as a function of time is going to be equal to M negative 0.25 times time plus plus 2 or if you want you could write it like this 2 minus 0.25 T actually like this form a little bit better it kind of in my brain it kind of describes what's happening a little bit more when time is equal to 0 you're starting it you're starting it to meter stick and then every week that goes by is T increases by 1 you're going to lose a quarter of a meter you're going to lose lose you have a negative value right over here you know you're gonna lose 0.25 meters and if you really want to kind of get this even in deeper level I encourage you to graph it it'll become even clearer what's going on here that this right over here is this right over here this is the slope of the of the the the line that represents the solution set to this equation and this to this would be your vertical intercept in this case would be your s intercept as opposed to your y-intercept when y is the vertical axis