Constructing linear models for real-world relationships
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- [Voiceover] "A lake near the Arctic Circle is covered "by a 2-meter-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 3 weeks, the sheet is only 1.25 meters thick. "After 3 weeks, the sheet is only 1.25 meters thick. "Let S(t) denote the ice sheet's thickness S "(measured in meters) as a function "of time (measured in weeks). "Write the function's formula." Alright, so 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 S of zero is equal to two. And they also tell us that after three weeks, the sheet is only 1.25 meters thick. And when we have the function S of t, S is measured in meters, time is measured in weeks. So after zero weeks, we're two meters thick, and then they tell us, after three weeks-- So S of three. After three weeks, we're 1.25 meters thick. Or another way to think about it, I could write t here in weeks and S in meters, and when time is zero, we're two meters thick, and when time is 1.-- Sorry, when time is three weeks, we are 1.25 meters thick. So when our change in time is equal to positive three, we increased our time by three, what's our change in thickness? Our change in thickness, the triangle here, that's the Greek letter delta, shorthand for "change in," well, this was negative 0.75. So what was the rate of change over this time? And they tell us that the rate of change is at 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 between any two periods of time, between zero week and one week, or one week and two weeks, or 1 1/2 weeks and 1.6 weeks. So 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? Well, we saw right over here. Our thickness went down, set 0.75 meters in three weeks. Or we could say that this right over here is equal to, let's see. 75 divided by three is 25, so 0.75 divided by three is 0.25. We have the negative out there, negative 0.25 meters 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 could 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 written as slope-intercept form. This is when you're dealing with x as the, I guess you could say the independent variable, y as the dependent variable, and b would be where you start. What happens when x equals zero and m is your rate of change, it's your slope? So in this case, 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? Well, one way to think about it is, well, what's S of zero going to be? S 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 two. And what is m? Well, we've already said, that's our rate of change, that is our slope, that is 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 zero comma two, and the point three comma 1.25, if we were plotting these points on a t/S coordinate plane. So now we can write what the function's going to be. 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 two. Or if you want, you could write it like this, two minus 0.25t. I actually like this form a little bit better. In my brain, it kind of describes what's happening a little bit more. When time is equal to zero, you're starting at two meters thick, and then every week that goes by, as t increases by one, you're going to lose a quarter of a meter. You're going to lose, you have a negative value right over here, you're gonna lose 0.25 meters. And if you really want to kind of get this even in a deeper level, I encourage you to graph it, and it'll become even clearer what's going on here. That this right over here is this right over here, is this right over here, this is the slope of the line that represents the solution set to this equation, and this two, this would be your vertical intercept. In this case, it would be your S-intercept as opposed to your y-intercept, when y is the vertical axis.