19 talks about a rectangular container so the container will look something of the sort. Wow, let's, uh, market markets dimensions. So it's X, y and Z. The volume of the container will be X y Z, and the volume has already given us 3 20. So here we have X y Z s 3. 20. So here the value off value of Z would be 3 20 over ex wife. Let's keep this. Let's keep it. Keep it aside. We're going to use it in the eventual calculations. Now we're supposed thio. The cost for the base has given the cost to make the basis $5 for food area. Off the base will be X times y. So it's cost. Let's say C one x and Y will be. Since that is $5 per square feature, it will be five expire on. We have the area off the sides as we have two sides venture from the richer towards the left and right as Y and Z dimensions to the area will be two y z plus. We have the front and back, which is two x z, so the total area would be two times y Z plus X z Z can be taken out, so it is to see times five or six. So the cost C two off the sides will be its $4 per square feet, so that will be four times this value. So it will be a Z Viper sex on Let's let's Reza substitute the value of Z as 3 20 over X plus y. So that is eight times 3 20 over to your new were ex wife. So it's x Y over here on board were here we have five plus x, so we have 3 20 times eight. So that is equal to 560 and the open of the brackets. We have one over X plus one or why this is C two X, so the total cost would be see even proceed to illustrate the cost function, which is five expire plus 2 +5601 of our X plus one of our while. Let's find C X now to maximize this so we have five I plus 2 +560 minus one over extra square and let's find see why that will be five X minus +2560 over. Why Square? Let's equate both these +20 So we have five y is equal to 2560 over extra square on. We have five x as 2560 over. Why square? So from here we get the value of y as this is 256 0/5 X square. Let's put this value off. Why? Over here? So we have five x as 2560 ver y square. But just 256 0/5 X square, whole square. So this comes out as five x is equal to if we open up the denominator than this stone will come up. So we have 25 x rays to the powerful since that is a square and we have 2560 square. So run off the stone will cancel our we have five xs 25 x rays to the poor, 4/2 560 So from here we have five times five is 25 let's cross multiply. So we have 2560 Xs five x rays to the par four mhm dividing both sides by five. We get 512 x as X rays to the poor. Four. This means that X rays to the poor. Four minus 512 x zero Let's take X out, X Q minus 512 0. This means that either x zero, which is just not possible because the land cannot be zero and X Cube is 412. Uh, it means that the cube root off 512 is actually eight. So exquisite fixes eight. We gotta find corresponding why which is over here. So why would be to 56 year over five times eight square. So 256 0/5 years. 512 and 512 divided by 60 fours again. Eight. Okay, so X is it? And why is it? And there is no point off calculating whether there is a maximum minimum because there is only critical point and the question asks about having a minimum, so the dimensions of the container would be access it. Why is it on the Z would be from here? Uh, from this equation, the Z would be 3. 20/8 times eight so 3. 20/64 as nothing but five, So the value of Z would be five. So these other dimensions for the container toe have the minimum cost.