For this problem. We are examining a sphere, so let's begin by drawing a picture. This is a three dimensional sphere. What do we know about this fear? Well, a sphere has a radius. Now, if the radius is constant, we could put a number on here. If the radius is changing, we're going to need to put a variable fee only numbers on our picture, our numbers that air unchanging. And we can see that the radius is of the sphere is expanding. In fact, it says it's expanding at a rate of 30 centimeters per minute. So since it's expanding, we're going to call it a variable. We'll call it our and I can write down that my radius is changing with respect to time. It's changing at a rate of 30 centimeters per minute. Okay? And we're looking at the volume. I want to know how the volume is changing with respect to time. That means what I'm looking for. It's the change of volume with respect to time. This is what I'm trying to find. So how can we relate the volume to our radius? Well, the volume of a sphere is four thirds pi are cute. So in order to find the change of volume, I need to take the derivative with respect to time. So on the left hand side, taking the derivative of V gives me d v d t On the right hand side I will take the derivative again multiplied by three four pi r That three comes down to a two and then I have to multiply this by the derivative of our with respect to t. So that's how all of these go together. Well, for this particular case, I know d r d t. That's 30. I know I want to find this. So my last piece is what is the radius? And if you read the problem, we're told that the radius at the point of time we're interested in is 15 centimeters. So I can plug in all of my values What I want to find d v d. T equals four times pi times 15 times the change of my radius, which is 30 centimeters per minute. And if I multiply all of that out, that gives me a value of before I do that. I realized I left off my squared, so it's four pi r squared d R D t So four pi times 15 squared times 30. That gives me 27,000 times pi. Or if I put that in a calculator, it is approximately 84,000 823 0.6. So that is how my volume is changing with respect to time. And if I want my units, I am in centimeters, so volume will be Centimeters Cube and my time is per minute so this will be centimeters cube per minute.