A farmer wants to fence in an area of $ 1.5 $ million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence?
So any time we're going to do any kind of optimization problem, especially if it's like geometric. In some sense, I like to draw a picture. So the way they kind of have this set up is so they say they have some fencing and then they're going to run a fence in between like the fencing in the middle kind of parallel. Like how I have this drawn here and what we want to do is find some way how we can minimize this cost while having a, um, fencing of area on the inside here of 1.5 million square feet. Actually, let's just go ahead and write this out without the whole million things. So just be 150 Okay, so we have this year. Well, if we were to think about this now, we're essentially just wanting to minimize our perimeter of this. So now I'll call these X is here, and then I'll call just like the top and bottom. Why? So our perimeter right now of this in total is going to be three x plus two y. But since this is in two variables, remember, we're gonna want to find some way to rewrite this. So we just have one variable so we can take the derivative. And over here, this area we're working with should be equal to Well, it's just the side length of the outer rectangle times the other side length of it. So just be X times why? And so I'll just go ahead and solve for why? So that would give us why is equal to 1.5 million over X, and then we can come over here and plug this in for why? Okay, so we're gonna end up with So p is equal to three x plus the two times that. So that's going to be three million now all over X. And now we can go ahead and take the derivative of this since this is in terms of Onley X now, So this is our perimeter function in terms of X on. This is assuming, like the cost doesn't really matter, since we're using the same kind of fencing for everything s okay. If we just use less of it overall, then we should just spend less is well, yeah, so now we want to try to take the derivative of this so we can try to minimize the function. Um, so the first thing we need to remember this is really here X to the negative one. So when we take the derivative of that so p prime of X, that would give us so three and minus three million over now X squared because power will remember says, take this move into the front. Subtract went off and what we want to do is set this equal to zero. So I'll go ahead and multiply everything by exc word. So it would be three. X squared is equal to three x word minus three million. So you got zero. Then we can add three million over divide by three. So that gives us X is equal to one million. Yeah, that's correct. Number of zeros. And then we could take the square root on each side, and so that would give us X is equal to plus or minus 1000. But in this case, it wouldn't make sense for us to talk about a negative side link. So we just throw that out and we have the positive one now. Uh huh. Um, and at this point, we can go ahead and plug this in to one of our equation so we could figure out what the other side length is going to be. Right? So, um, over here we can just plug that into our constraint equation. So we now take this plug in 1000 there. So give us why is equal to, uh, 1.5 million divided by 1000. So essentially just canceled three of these zeros for each. And so then that tells us why is equal to 1500 opened. Both of these should be in feet, by the way. So if we were to kind of draw this again and look at what are dimensions should be so we should have where our top ones. So going across here like this is going to be 1500 ft, and then the three sides are going to be 1000. Because if we come up here and look at this again, it was x x x and then why? Why? So I guess we could kind of write this out. So we want three sides with 1000 eat and two sides with 1500 ft, so these would be the dimensions we would need in order for us to minimize our total cost