 In this video, we provide the solution to question number six for practice exam number four for math 1210, in which case we're asked to find the minimum perimeter of a rectangle with an area of 100 square centimeters. So perimeter is gonna be your optimizing function here. So perimeter beware is equal to two times length plus two times the width. We wanna minimize this. What do we know about the area? Well, we know that area is length times width and it equals 100. So we can solve for one of these variables. The length is a little bit harder to draw because it requires cursive here. So we get that the length is equal to 100 over W. So we're gonna make that substitution in over here so that the perimeter is equal to two times L with L is 100 over W. And then we're gonna get two times W right here for which we can take two times 100. We end up with 200 times W to the negative one plus two W. I changed it from a fraction to a power to prepare ourselves to compute the derivative as we do these optimization problems. If you calculate the derivative, you're gonna get that P prime is equal to negative 200 W to the negative two plus two. And so we have to figure out when this thing is equal to zero. It is undefined at W equals zero, but if a rectangle where the width was equal to zero, that would be impossible with our constraints. So we don't have to worry about that one. We have to figure when this thing equals zero. So I'm gonna add 200 W to negative two to both sides. So we get two equals 200 W to the negative two. Divide both sides by 200. This is gonna give you two over 200 or in other words, one over 100. This is equal to W to the negative two. But what does the negative power mean? It means one over W squared. So taking the reciprocal, we get W squared equals 10, excuse me, 100. Taking the square root, we get that W equals the square root of 100, which is equal to 10. Now we have to be very careful here. One of the responses is 10, but what is the question asking? Does that question ask for what width should we have to minimize distance, or to minimize perimeter or anything like that? No, it asks for what is the minimum perimeter? So if the width is 10, what is the length? Well, the length would be 100 over 10, which gives you 10 itself. So it turns out that the minimum perimeter will happen with a square, not just any old rectangle. It's gonna be a 10 by 10 square. You see this a lot when it comes to optimization. It seems that things are optimized when they are symmetric. Now I don't want you to take that too literally because we don't always understand what symmetry means in every situation. But in this situation, turns out the minimum perimeter does happen at a square. Maximum area would also happen at a square. And so the perimeter of the square will be 10 plus 10 plus 10 plus 10, which is 40. So actually the correct answer is A, 40 would be the minimum perimeter.