 This video will be applications of exponential functions. So Newton's law of cooling is a very important law, especially if you're talking about science. And T of x is equal to T sub r plus blah, blah, blah, blah. T of 0 is the initial temperature of whatever we have that we want to know its temperature. T of r is the temperature of the room or the surrounding medium. T of x is going to be the temperature of the object x minutes later, and k is a cooling rate. So we have a pot of coffee with a temperature of 100 degrees Celsius, and it's set down in a room with a temperature of 20 degrees Celsius. And we don't want to do this because we're actually going to find that, and it's not that value. So we want to know what that cooling rate is. So we're looking for k, and the 60 degrees Celsius is going to be the temperature. So that's T of x, and then this one hour is telling me that x is equal to 60. We'll have to talk about 60 minutes because up here it says minutes later. So we also have this 100 degrees and 20 degrees. Well, what are those? The initial temperature was 100 degrees, and then it's going to be in a room temperature that's 20 degrees, so this would be T of r. So it's right, put in what we know. T of x is 60 equal to T of r, which we said was 20, plus the difference between temperature and the room, so 100 minus 20, times e to the k times 60, because it has to be 60 minutes. We talked about that. So if I bring the 20 over here, I'm going to have 40 is equal to, and if I subtract here, that'll be 80 times e to the 60 k. But I want to get my base and this exponent all by itself, so I need to divide off the 80. So 40 divided by 80 is going to be 1 half is equal to e to the 60 k. And now we're ready to really start solving this. So when we solve exponential functions like this one, e is not a nice number. It's a decimal number. At this point, we don't really know what else to do since we have this unknown up here in the exponent. So we need to graph or do something else. So let's graph this one. Put it over here, go to y equal, and I want to put 1 half or 0.5 for one equation. This is y1 down here. And this would be y2. And we put in clear y2. And we have second ln. Remember that's how you get to e. And then in the parentheses, we have 60x. And we can do a couple of things. We can graph or we could look at our table. Graph, if it's a decimal answer, graphing will probably help us out a little more. And we can't see our other graph. So let's change our window. Let's let this go from, say, negative 2 to 2. And we want to go every 0.001. And let's let this go from negative 5 to 5. And we want to go every 0.001. And see what our graph looks like now. There's our 1 half. And there's our exponential. It's so small, it's in here, but we could try second trace now. So we could see something in there. Enter, enter, enter. And it's going to give us negative 0.015. And we're going to say that k is equal to 0.012. Negative 1.012. So we need to keep this in mind when it says, what was the temperature of the coffee after 30 minutes? Let's just change colors here and do it right here in the middle. So now I know a t again. And I want to know t of x. So t of x is going to be equal to, we can use a simplified formula. 20 plus 80 times e to the now I know k. Negative 0.012 times 30 minutes. We use 60 minutes for an hour, 30 minutes for this half hour. And if we plug and chug all that in our calculators, when I know the temperature, the temperature is 75.8 after we plug into degrees Celsius. And then next we have, when did it reach 50 degrees Celsius? Well, we know that's t of x. And we know that it was 20 plus 80 times e to the negative 0.012. That was our k. And then we want to know the time. So we subtract 20, we'll have 30. Equal to 80 times e to the negative 0.012. Divide, we'll have three over eight is equal to e to the negative 0.012 t. Go back to our calculator. We have e to the, we need to change all of these. So on this side, we want to say three divided by eight. And then we want to clear this one out. And we want to say second l n negative 0.012 times x. I guess I didn't need the times. And we need to change our window. So not zoom. We need to change our window. We'll say that we want to go from zero. Because this is when, so time, when time begins. And let's say we'll go to 100. 110 is okay. Every one. And we can go from, we can try negative five to five. But let's go every one. And if we look at a graph, there's our three eighths. And here comes our exponential. And if we second trace five, enter, enter, enter is the intersection. We find out it takes basically 81.7 minutes. So t is equal to 81.7 minutes. And we had a graph that looks something like this. With this coming down here like this. And that's what we got. All right. So photocopiers have become a critical part of operations of many businesses. And they've got a depreciation equation as v equal v zero, five eighths over t. v zero is the initial value. t is in years. And we want to know how much it is after two years. So how much is a copier worth after one year? So that's t. If it costs 64,000 new, so that would be our v zero. We want to know the value. If we have 64 to start out with, we take our five eighths base and we raise it to the first. And if you take that to your calculator, which I would like for you to do at this moment, then you get 40 or $40,000. That's how much it's worse after one year. And how many years will it take the copier to depreciate to 25,000? Well, now we know the v. I'm going to work up here. It still started out at 64,000 and our base is still five eighths. But now we don't know t. So we're going to have 25 divided by 64 is equal to five over eight t. And it looks like we can get the same basis here. So this would be five squared over eight squared is equal to five over eight to the t, which is really going to just be two equal t. Because this one would really be five over eight squared would be the same thing as five squared over eight squared. So t is equal to two. All right. If a company's revenue grows at a rate of 150% per year, the revenue is given by this equation. Three over two is 1.5. So 150% is right there. R zero is the initial revenue. R t is after t years. So how much revenue is being generated after three years? So that's t. If the company's initial revenue was 256,000. So that's R zero. So R of three, because we know that t is three, will be 256 times three over two. And we know t to be three. So that's just a plug and chug. And if you plug and chug all that wonderful stuff, you're going to find out that that is $864,000. How long will the company be until it's generating 1,944? Well, that's R. We're going to have to solve for t. So 256 times three over two, the third, we're going to have to take that in 1944 and divide it by 256. And then I want to convert it into a fraction to see if nice things happen for us. And it doesn't really, it doesn't take us down to something like three over two. So we'll have to graph when we solve this one. But this isn't three. This is t. Okay. So my program is doing some funny things so I can't write anymore. But we're so close. I'm just going to have them finished. So this three over here should be a t. And then I've put the two sides of that equation. I divided by 256. So there's my 144 divided by 256. And that leaves me with my three halves to the x. That's my second equation. And then I graft it and I just use the standard window. And I can see my intersection. So second trace five would be the intersection. And then enter, enter, enter five years for the revenue to be that or 1,944. Thank you.