 In this video, practice problem three. We're given the surface area of a cone and we're given the slant height and we're asked to find the radius. So if you don't know where to start on these, we start with the formula. We know that we're talking about the surface area of a cone, so I brought up the formula. Surface area equals pi RL plus pi R squared. And let's look at the information given to us. If we're given the total surface area is 39 pi, we know that this is one of those problems where we're going to have to put in the information we know and work backwards. First, I like to look at what we're asked to find. We're asked to find the radius. I'm just going to circle that in my formula to remind me of what I'm trying to find. And now I'm going to put in the information I know. I know the total surface area is 39 pi. This represents the total surface area. And I know that the slant height is 10. Remember, the slant height is represented by the curse of L. So let's just plug that into our formula. We know we're trying to find R, so that's going to stay there. The slant height is 10, so that pi RL becomes pi R times 10 plus pi R squared. The pi's have to stay in there because the pi's do represent a number. And let's just clean this up and see what we have. 39 pi equals, I'm just going to rewrite this term as 10 pi R plus pi R squared. And now we have an equation where the only unknown variables here are R, and that is what we're trying to find. So let's see if we can clean this up. We see that we have three terms here, 39 pi, 10 pi R, and pi R squared. What is common to each of those three terms is of course pi. Each of them has pi in it, so I can cancel that out. And then I want to also look and see if I can cancel out a number. This first term has 39 pi, 10 pi, and then there's nothing on this last term, that's just a 1. So 1, 10, and 39, I can't cancel anything out of there, so I'm just going to go ahead and cancel the pi and see what I have. So when I divide this term by pi, those cancel out and I'm just left with a 39. Cancel out the pi here, I'm left with 10 R, and cancel out the pi here, and I'm left with positive R squared. And you can see when I clean this up now, I have an R squared and I have an R term and a number term. Any time you see that R squared, that should be a tip-off that we're going to put that in standard form and factor it. Putting it in standard form just means I'm going to move that 39 over, put it in standard form, R squared first, and set everything equal to 0. And this is where I know when I factor it, I know it's going to be R and R. And then I have to look at this number, find the factor pairs of 39, I'm going to just write over here. We only have two sets of factor pairs of 39, 1 times 39, or 3 times 13. And we have to look at these two separate sets of factor pairs to decide which of them will add or subtract up to get us to the 10. And we know that 3 and 13 will get us to the 10. So I'm going to start by just writing the 3 and 13 before I talk about the signs here. If I want to get a positive 10, I know I need a positive 13 and a negative 3. And I like to do that on the side and then I can come over to my solution and put positive 13 and negative 3. And that will help, hopefully, reduce some mistakes. And then my final step then, of course, to solve for R, my R answer because that's R minus 3 would be positive 3 and negative 13. And this is where I have to look at what I'm actually trying to find. I'm trying to find the radius. That's what this R represents. If R represents the radius, yes, I can have a positive 3 radius, but I can't have a negative distance. So my final answer is just going to be the one of those, R radius equals 3 units. And remember sometimes that negative answer could work. You just have to look at what they're asking, see if you need to plug it in and see if that makes sense. But in this case, we just have the one answer.