We've recently been spending a lot of time focusing on related rates problems. Recall for these problems that we see, we're dealing with situations where we have an equation or a shape that is changing over time. It turns out that a lot of these related rates problems that you're going to see are going to be real-world application problems, which, in my opinion, are some of the trickiest problems to solve. But don't sweat it because in this video, I'm going to walk you through an example of a real-world application problem and show you the process for solving these types of problems. I think you're going to find it's very similar to the process we've used for dealing with related rates up to this point.
So let's just go ahead and get right into things. Now in this example down here, this would be an example of a real work problem. We're told as an ice cube melts, each side changes at negative 3 centimeters per minute. Find the rate of change of the ice cube's volume when each side is 0.9 centimeters. Now the way that you can tell you're dealing with a real-world situation here is because we have this ice cube melting, and we're not really given an explicit shape or equation that we need to solve for, and we have to extract all that information ourselves.
So, for real-world problems, you need to determine what shapes are formed and how they change over time. Now the way that I'm going to do this is just by walking through the same steps that we've seen in the past with related rates, which is to first start by drawing and labeling a picture of the scenario. Now all I really know about this situation is that I'm dealing with an ice cube that is melting. But what I can figure out is, based on the sentence here, what information and variables I actually have. We're told each side changes at negative 3 centimeters per minute.
Now I want you to pay close attention to something. Notice how each side is changing at negative 3 centimeters per minute. We were told in previous videos when dealing with a positive sign, that means your shape is growing. When dealing with a negative sign, your shape is shrinking. Since I can see that we have a negative sign here, that means that this entire ice cube is shrinking, which makes sense since it's a melting ice cube.
Now I'm going to focus on one side of the ice cube even though all sides are the same. I'm just going to focus on this side for now since I know that all the sides are going to be melting uniformly. Now what this side is doing is it's shrinking. It's becoming smaller at a rate of negative 3 centimeters for every minute. So that's one of the rates that we have.
Now I'm also told at one instance, this ice cube has each side equal to 0.9 centimeters. So I'll focus on another side of the ice cube here. Again, they're all the same. But I'll focus on another side here, which I'll say is 0.9 centimeters at one instant. Now what I'm trying to find is the rate at for the ice cube's volume.
And the way that I can do that is by recognizing since this shape is shrinking, the ice cube is melting, this entire volume is going to be shrinking, and that's the rate that we're ultimately looking for. So this is what we're trying to find. So what variables do we have in this situation? Well, I can see that we have the rate for which one side of the cube is changing. So that's going to be dxdt, which is negative 3 centimeters per minute.
That's a changing side with respect to time. Now I can also see here that we have one instance where the side of the cube is 0.9 centimeters. So each side is going to be this length at one instant. And what we're ultimately trying to find is the rate at which the volume is changing the whole cube, which is dvdt. So that's what we don't have.
Now based on these variables here, I've gone ahead and done step 1, which is labeling the situation and finding these variables. So my second step is going to be to identify all the equations or identify all the equations we need that relates these variables. And if I'm thinking about the equations that are going to work, the one equation that comes to mind is the volume of a cube. So you can see that we have an ice cube. All of these have the same length, that it's uniformly melting.
And I can see we have the rate for one side changing, and we're looking for the rate at which the volume changes. So I'm pretty sure the volume of a cube, which I know is just going to be one side x raised to the third power, I'm pretty sure this is going to give me what I'm looking for. But let's see if it does by doing this third step here, which is taking the derivative on both sides and using implicit differentiation. So I'm going to take the derivative with respect to time of this entire equation. So that means I'll have the derivative with respect to time of the volume, and that's going to be equal to the derivative with respect to time of this term right here, x cubed.
Now I can find the derivative of volume with respect to time, which we know using the chain rule is just going to come out to dvdt. And for finding the derivative of x cubed, well, using the power rule, I know that that's going to be 3x squared. Then I need to multiply this by the derivative of x itself using the chain rule since x is the inside term, and that's just going to give us dx over dt multiplied on the outside. So this right here is the equation that we need. And something I notice about this equation is this equation has dvdt, the missing rate.
It has one side of the cube, and it has the rate at which one side is changing. So this is indeed the equation that we needed to solve this problem. So from here, now that I've found this third step, my fourth step is going to be to isolate the target rate of change. Now the target rate of change is dvdt. That's what we don't have.
But notice how that's already on one side of the equal sign. So since we already have this isolated on one side, we can go ahead and skip this fourth step here because it's already solved for us. So the last step here is going to be to plug in the known values and rates and use that to solve this problem. So I'm going to do that over here. Now what I can see is that we have x in this equation.
That's going to be one side of the cube at an instant. Well, all sides are the same. So what this is going to mean is that all sides of the cube are going to be 0.9 centimeters at one moment, which is what we're going to plug in for x. So we're going to have 0.9, and then this whole thing is going to be squared since x is squared in this equation, and that's going to be multiplied by the rate dx over dt. And I can see that this rate is negative 3 centimeters per minute, so that means this whole thing is going to be negative 3.
Now what you can go ahead and do from here is multiply these numbers on a calculator, and if you plug this all in, you should get that your rate is equal to negative 4.86. So negative 4.86 is going to be your answer, and since we have a changing volume with respect to time, it's going to be in cubic centimeters per minute. Now notice that we got a negative result for the rate dvdt. But this actually makes sense because we're dealing with an ice cube that is shrinking. And since the whole ice cube is shrinking, that means the cube is shrinking, which means we're going to get a negative rate for the changing volume with respect to time.
So this is how you can solve some of these more complicated related rates problems, where rather than being given a shape or an equation of any kind, you're just given a scenario, and you have to extract all the information from that scenario. Notice it does take a little bit more effort with the setup, but the process is the same. So hope you found this video helpful, and let's go ahead and get some more practice with these types of problems.
