 A couple more examples involving volume of prisms and cylinders. In this example we want to find the volume of a rectangular prism that has had a hole drilled through it. And you'll notice that the hole happens to be a cylinder. So what we're going to do here is we're going to take the volume of the prism and we're going to subtract the volume of the cylinder. And what that will do is it'll give us this darker shaded region that's around the hole. So volume of the prism, let's start with that. The prism is a rectangular prism, which means that the bases are rectangles. So in order to find the volume of a rectangular prism, I'm going to take base times height of the rectangle times the height of the prism. If you look at your picture, you can see that the rectangle is a 6 by 5 rectangle. So 6 times 5. And you can see that the height of this prism is 4, so I'm going to multiply by 4. And then the volume of the cylinder is going to be pi r squared because that's the area of the base times the height of the cylinder. Well, if I look at the picture, it tells me up here that the hole has a radius of 2. So pi times 2 squared. And the height of the cylinder is the distance from the top to the bottom, which is exactly the same as the height of this prism. So it is a height of 4. So now if we multiply 6 times 5 times 4, we get 120. And in the second set of parentheses, 2 squared is 4 times 4 is 16 times pi is 16 pi. And this question didn't ask us to do exact or approximate. So if it asked to stay as an exact answer, we would just leave it as 120 minus 16 pi. If you want to put that in your calculator and approximate it, you would get about 69.7. And again, both of these should be labeled with cubic units. So just, again, remember that this is the exact answer and this one here would be the approximate answer. In this example, we have to work backwards. If you read what it says, it says that we're going to find the radius of the base of a cylinder if the volume is 126.7 cubic feet and the height is 10 feet. So basically what we're doing here is we are given the volume, we're given the height of the cylinder, and we want to figure out what the radius is. So remember that the volume of a cylinder is the area of the base, which is pi r squared times the height of the cylinder. So I'm going to plug in what I know. The volume is 126.7 and then I'm going to leave it as pi r squared because we don't know what r is and that is multiplied by 10, the height. Now this is just an equation that you're going to solve for r. So the first thing I'm going to do is get rid of the 10 and the pi and I can do that in one step because it's multiplication pi times 10. So if I divide by 10 pi, it's going to cancel out the 10 and the pi. I have to do exactly the same thing on the other side. Now I'll just tell you when you enter this in your calculator, be really careful. Make sure that you do 126.7 divided by 10 times pi, which is probably going to involve you using parentheses around the 10 pi. The other thing you could do is enter 10 times pi to get the decimal and then do 126.7 divided by that decimal. What you should end up happening or what should end up happening is you have r squared on the right side and if you do 126.7 divided by 10 pi, it's like 3.99. So we're going to go ahead and just round that to 4. And so the last step for solving for r is to do the square root and we get a radius of 2. And because this is in feet, we would say 2 feet.