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An open-top rectangular box is being constructed to hold volume of 200 in? . The base of the box is made from material costing 5 cents /in? The front of the box mus...

Question

An open-top rectangular box is being constructed to hold volume of 200 in? . The base of the box is made from material costing 5 cents /in? The front of the box must be decorated and will cost 10 cents/in2. The remainder of the sides will cost 2 cents/ in2.Find the dimensions that will minimize the cost of constructing this box:Front widthzDepth:Height:

An open-top rectangular box is being constructed to hold volume of 200 in? . The base of the box is made from material costing 5 cents /in? The front of the box must be decorated and will cost 10 cents/in2. The remainder of the sides will cost 2 cents/ in2. Find the dimensions that will minimize the cost of constructing this box: Front widthz Depth: Height:



Answers

A rectangular box with square base and top is to be made to contain 1250 cubic feet. The material for the base costs 35 cents per square foot, for the top 15 cents per square foot, and for the sides 20 cents per square foot. Find the dimensions that will minimize the cost of the box.

For this problem we are told that the cost of producing a rectangular boxes as follows. The sides cost $2 per square foot. The top $1 per square foot in the base. $4 per square foot were then told that if the volume of the boxes to be 10 cubic feet were asked to determine the dimensions that minimize the cost to begin. We'll have that. The cost will equal. Now we have the sides will be um One it is $2 per square foot. For the sides. We have $2 times. Now we have each side will be You have two times X. Y. for the front and back. You'll have two times Y. Z. For the other two sides. So we have two times two X. Y. Plus too easy for the sides. Then we have plus one times the cost of the top. The top is going to be X. Times E. And also we have that the base is $4 per square foot. So that will add on another four exit or plus five exit. So now we have our costs, we can multiply that too in so it's four X. Y. Plus four Y. Zed Plus five X. Set. And we're told that the volume must be 10 cubic feet. The volume is going to be X. Times Y. Times Z. And it must equal 10. So what we have to do here is figure out a way of minimizing the cost subject to the constraint X. Y. Z equals 10. So what we can do here is use the method of lagrange multipliers. So the first step is that we want to set the gradient of c. Radiant of C equal to lambda times the gradient of V. So we'll have first, when we take the partial derivative with respect to X, we'll have four Y plus five Z equals lambda times wise. It Then we have with respect to why we have four x plus four said equals lambda, X. Said. We then have uh with respect to Z. We have four Y plus five X equals lambda X. Y. And we also have our constraint X. Y. Z. equals 10. So we have a system of four equations with four unknowns one second here. So the solution that we get from that system of equations will be lambda equals for x equals two. Y equals five over to n. Z equals two.

We want to find the dimensions of this close box such that to get maximum volume With the daughter cost kept $240. The top, The cost is $20 per meter square and the such And the base is a $10 per meter square. So for maximum volume we want to find the dimension. And so let's name the side X. And decide why. And we know that this is one m. So it's logical to create a cost equation. I will be taking a top area multiplied by $20 collusion plus the base area. And the sites area times $10. Now for the top, the area will be one times X. So just be x times 20 class for the basis. Also one X. So it's just X. Plus the site will be, there are two personal sites for the first pay over here. The area is one Y. But there are two of them in front and the back. So it would just be to Y. plus the other site would be over here, this side and the other side, the opposite side here. And that would be X. Y. Times too. So it's two X. Y. And this whole thing is multiplied to $10. Okay so we are simplifying this to 20 x. Last 10 X. Last 20. Why? Plus 20 X. Y. I'm gonna set it to $240. As you can see here. I can just Do you buy everything by 10? So I'm just going to get rid of all the zeros here and so on the left I can write my two X plus two. Why? Where I'm combining my my wife together here by writing you on the left is equals two 20 for -3 x. So my wife is equal to 20 for Ministry X. Over two X plus one. Now since we won the to find maximum volume or the dimensions such that we get maximum volume. We should do the volume volume equation. So volume east one times X. Times Y. So it's just X. Y. So The Y. For this one 24 minus she X. Over to express one. Now it's easier to multiply the X. In. So easier to your friend shit. So let's differentiate with respect to X. Yeah using caution rule I can leave my half outside losing Kocian rule. You're going to square the denominator, repeat the original denominator at the top, differentiate the numerator. You'll get Now 24 X. We differentiate you get 24 and minus three X. Square. When you differentiate you just get minus six X. two A- Sign. Now we freeze the new America. Okay And we're going to um we're gonna we're going to differentiate the denominator and that's just gonna be one. Okay, I'm just running out of space here. Yes, she's square here. Now when you tidy up this is what you will get. You get minus tree, X square minus six X plus 24 over X plus one. Okay. Square. All right now we're gonna set DVD X equals to zero. Now when you say it goes to zero only the numerator is equal to zero. Denominator cannot be equal to zero because you cannot divide by zero. So we have ministry X squared minus six. X plus 24. 0. You will get X Equals to two or X equals 2 -4. You will reject the -4 as dimension is positive. Now when actually goes to stop it into here, two other ex here to get your y. So your wife is why is tree. So therefore the dimension for maximum volume will be one Thumbs, two Thumbs 3 m

Okay We want to find the dimensions for this box such that the volumes maximum but kept at the cost of $240 to make it for the top. It's $20 per meter square to make it. And for the sex and the base is $10 per meter square. So it's logical for us to create a consecration Where I'll be taking a top area multiplied by $20 plus the base area. And it's areas What applied by $10. Now the top. Now let's the dimension for this site, B. X. And for the high B. Y. Now for the top, the area will be one X. So let's just be X. Times the $20 now plus. Mhm. These areas also one X. Was just X. Plus. Now there are two possible site for the first site is the front and the back. It will be one. Why? So it's just going to be For two sides there'll be just two Y. Plus. Now for the other side this site and the opposite side here is X. Y. And there are two of them. So it's two X. Y. And multiply by 10. So when I woke it up I will get 20 x plus 10 x plus 20 Y. Just 20 x. Y. Now I'm gonna set $240 to it And I can divide by 10. So all the zeros are gone. Let's move all the Why to one side and the rest of the other side. So I'm gonna write that my wife on the left, this is what I will get And it will be 24- tricks. So my wife is 20 for minus three X. Over two X plus one. Okay, so now the next thing to do is to create a volume equation. Now volume will be one times X. Times Y. So just be X. Y. Stop at the Y. With 24 -3 x. Over two x plus one multiply the X. Into the bracket. So it's easier to differentiate later. Yeah Using caution rules and we differentiate DVD X can leave the half outside square. The denominator repeat the original denominator at the top, differentiate the new morita. And we're differentiating Marita 24 X. will become 24 minus not three X squared when your friendship I will get six x minus no freeze the numerator. Yeah in different shape the denominator which is just X plus one. So when differentiated I'm just going to get one. Okay now we can tidy up and after you tidy up this is what you will get. Okay we're gonna set DVD X 20 because we want to find maximum volume so find it at the stationary point When we set this to zero. Yes the numerator is equal to zero. Not the denominator. As denominator cannot be zero. You cannot divide by zero. So I will get minus tree X square minus six X plus 24 because 20 and that gives me X. Is equals two. Hello two Or x equals 2 -4. Now we will reject the negative as dementia is positive. Now we're supposed to we're gonna suck the excuse to do into here to find a. Why? So the Y. Will be tree. Therefore to get maximum volume Where the cost is kept at $240. My dimension would be One times two times 3 m. Yeah.

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.


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