So here we have a closed box where the base is a square. Please let's go ahead and draw that box. He said it'll look something like this. Mhm. Yeah. So now we're going to say um so let's say that since the basis of square, we can call that X. That's also X. And then the height will just call that white. So we have this box with a given volume. Okay? So which means we don't know what the volume is, but it's a constant, right? So because the volume has to stay constant and then we can kind of stretch the blogs make it shorter, make it taller. However we want as long as the volume stays the same. So we'll say that the volume which is a V. Is going to be the base which is X. Squared times the height. Which is why. Uh huh. So what we're trying to do is we're trying to minimize the cost of making this box. So what we know is that the material used in the bottom costs 20 more percent um than the material in the site's okay. And then the top costs 54%. So what we can do is we can say, okay, assume that the cost of of the sites is just one, okay? Not necessarily $1 but just proportional wise. That's what and the classroom where the bottom, the box is going to be 1.2 and the cost of the top of the box is going to be 1.5. So therefore, if we have a cost function, this is going to be, so the cost of the bottom, that's one point two times the area of the bottom, which is X squared plus the cost of the sides. So each side is a rectangle with a base of X and a height of one. So that would be four X. Y. And the cost of the side we're saying that's just one. And now plus the cost of the top, which is going to be 1.5 times X squared. Okay. So this is what we're trying to um trying to minimize. Okay. So what we're going to do is we're actually going to use this volume equation to isolate why. Okay. So that we can get it in terms of X. So if we do that, that tells us why is actually equal to V. X. To the power of negative two. So now we can substitute that back into our cost function. And we get this is so we can combine the two. So that's 27 X squared plus four X times the extra power of negative two. So there's 27 X squared plus for B. X. The power of negative one. All right. So now if we're talking about the domain then X. Y both have to be greater than zero. Or else we won't have the box. Okay? So which means um this is okay right here because we will never be divided by zero. So now we can go ahead and find a critical values. So we do that by taking C prime, which is not only a function of X. So this is going to be 5.4 times X. And this this will be minus four V. The path times except power of negative two. Which is really just for V over X squared. Okay, So to find the critical values will set this equal to zero and then we try to solve for X. It's for me. Okay? So let's move this over. We get four V over X squared is equal to 5.4 X. So four V over 5.4 is equal to X. Q. So therefore X is equal to four. The over 5.4 to the power of one over three. Okay? So now um we've basically found are critical value. Okay? So this critical value is going to give us the minimum. Okay? And you can do that, you can make sure that the minimum by doing your secondary test. So in fact let's go ahead and do that. So see double prime is going to be equal to 54 and this will be plus eight V over execute. Okay? So if we substitute this value and so see double prime of four V over 5.4. All to the power of 1/3. We get this is 54 plus eight V over four V over 5.4. So we can actually multiply by the reciprocal. That's a week, times five point 4/4 B. That will cancel this becomes A two. And we can see that this is greater than zero. So if it's greater than zero and that means this is going to be concave up which means are critical value is in fact a minimum. Okay, so uh let's see what the question is actually asking for. It's asking us to find the most economical proportions. Okay. So we want to find the proportions. So therefore we would say in this case the proportion is X is going to be equal to this and we would also go ahead and find out why. So why is going to be the divided by um X squared? So that would be for the 5.4 the power of to over three. Okay, so this would be our proportion.