an-open-box-with-locking-tabs-is-to-be-made-from-a-square-piece-of-material-24-inches-on-a-side-this

Question

An  open  box  with  locking  tabs is  to  be  made  from  a  square  piece  of  material  24  inches on  a  side.  This  is  to  be  done  by  cutting  equal  squares from  the  corners  and  folding  along  the  dashed  lines shown  in  the  figure.

(a)  Write  a  function $ V(x) $ that  represents  the  volume  of the  box.
(b)  Determine  the  domain  of  the  function $ V $.
(c)  Sketch  a  graph  of  the  function  and  estimate  the value  of  for  which $ V(x) $ is  maximum.

Sheryl Ezze · Accepted Answer

So you have that picture shown to you on how they have this 24 by 24 inch, uh, square piece of metal. And interestingly enough, this problem was actually new to me. I hadn&#x27;t seen this before. That there, they&#x27;re going to have thes corners. Traditionally, we just deal with these corners being axed, but they show on there is how they&#x27;re actually gonna fold over the tab so that they make the box dirtier. So there, actually, the box they haven&#x27;t shown like this. That and I&#x27;m not going to draw it folded up. But this is the bottom of the box. And so the bottom of the box is gonna end up being longer than it is going to be wide. And so we know that the box will end up looking like this and again, those tabs, they&#x27;re going to be folded over. So we have to tell what these dimensions are. And, yes, this is going to be ex tall now on this side. This part of the box from here to here, which I&#x27;m showing from here to here, is that 24 inches minus two exes. However, this side of the box which I have shown right here is going to be the 24 inches minus 1234 axes. So the volume of that box, which is what we have to answer in part a is X Times 24 minus four X Times 24 minus two X. Right. And, uh, I would let my students leaving in this form, but they do show the answer as all expanded out. So let&#x27;s quit, multiply. We multiply the 24 times 24 24 squared whips. Let me clear out my calculator and get out of a mode that I didn&#x27;t want to be in. We get 5, 576 and then we&#x27;ll have right here. We&#x27;ll have minus 48 X and then right here we&#x27;ll have 24 times four. That&#x27;s giving us minus 96 X and then multiplying. Here we get plus eight x squared and now we&#x27;ll combine terms and distribute that X through. So this will be our polynomial. So x times eight that will give us that eight X to the third power. Well, the 96 plus the 48 or the both of them are negative is negative. 144 x and then this is gonna be X squared. We distribute the X through and then 576 X and so we can put that expression into our calculator. And so let me just quick put this in and, um, on subtle hint, you can put in this expression as wise of one and then you can also put in the other expression as wise and to, and you can double check your multiplication. So, um, I almost have that typed in here. Almost done. Just a second. Just a second. And I&#x27;m gonna check my multiplication so I put in. This is why I think this one is why someone and this one is Why said to and now I&#x27;m going to do a table of values now. What kind of values can we have? Well, let&#x27;s look at this box. We can see that X naturally has to be bigger than zero. Otherwise, we&#x27;re not gonna be, uh, holding it up and have any any height. And this is going to be the part that is going thio affect our expression the most. If we had XB six, we would have this side only be zero long, so it has to be less than six inches. So that cut X has to be less than six inches. So now when I go to my table set, I&#x27;m gonna start my table at zero, and I&#x27;m going to go up by ones. And when I do, I&#x27;m just going to kind of look through the values and I can see that the highest looks like it&#x27;s at three inches. At three inches, it&#x27;s all the way up to 648. So I&#x27;m gonna set my graft to graft this. I&#x27;m going to go from 0 to 6. My ex access and my window is going to be from 0 to 6 and I&#x27;ll scale by ones. And then I&#x27;m going to go up to I&#x27;ll go up to 700 on this axis, so I&#x27;ll have my wine minimum B zero. I&#x27;ll go up to 700 scale by hundreds, and when I dio, I get this graph. It goes up and it&#x27;s a little bit skewed and it comes down and I wanna find that maximum. So we&#x27;re gonna do our second calculate and we&#x27;re gonna find this maximum point. And it happens to be four on my calculator. I&#x27;m going to get to the left of it and enter. And then I&#x27;m going to get to the right of the maximum and enter, and then I&#x27;m going to add it and just get my guests. And it tells me that at 2.5 36 inches, we have the maximum volume being 665.1 one. Uh, and that would be Cuba tensions. So that is that maximum point?

Ankit Gupta · Answer

we have given a question and open box with locking taps is to be made from a square piece off material 24 inches on the sides. This is done by cutting equals square from the corner and folding along their deaths is lying as shown in the figure we have given him Figure in the bushes. And first we need to tell that this last portion is a question is verified that this volume, our boxes given by function we have given a function off this morning. We excessive will do it X and do six minus X in tow. 12 minus X, and we need to very fighters so we can solve this equation. As as we can tie this one as doing toe turn minus X in tow, more into six minus X into x No, we can solve this one as 24 minus two x into 24 minus two x into its No. We know that volume is equal to lend a little bit into height. So from this, we can easily right That he&#x27;ll meant is a little 24 minus two x after comparing the desecration. Really, it is 24 minus two x and hi days Hi, it is X here the civilians four x section of the full moon bullets. So this is not flushed on. So And for a second answer, relatable for no man the neutral right moment for this volume. So for that. So I didn&#x27;t mind the boiling for this letter. Mining dormant for this volume. We can cut something. We can guard this one. We need dough. Good. Something so X is greater than a little. But we can&#x27;t, Kurt. More than off the side. Lend. Oh Materia, We need to get something, actually but we need to remember this big. Aren&#x27;t card more than half of the site, so we can diet to you. Two x iss 24 by six is in Quito. So this is the moment on here, recondite exes? No thanks. No for C. Push it toward Cushion Many Tuesday afternoon. Energy on that. We need to the schedule for this. We need to have Grandfather&#x27;s after using graphing ability we ever got for this. So this is a graph from graphing utility here. We can see that the volume is maximum at this point, that just at the point off 2.5. We can write this one the volume maximum maximum or cause and the boat accessing nearly about 2.5. INGE&#x27;s volume is maximum at two, so it is maximum or correct access near neighbour dupont cabbage.

Ankit Gupta · Answer

we have given a question and open boxes Toby made from us where peace off Materials 46 images on a side by cutting equally scores with side off, Lend X from the corners and turning up the sites and first pushin Ezio verify that the world live off the boxes given by the function here given for the flashed pushing function is re X is equal x 36 point this two x spread and we need to verify this is a limb or not. So first we can write this one as X in tow 36 minus two x and it is a square. So we going to feed this toe 36 minus two x on We contrite this one as 26 minus two weeks 36 minus two x and X and we know that volume we live is equal toe lend into with Indo height. So from this equation, we need to compare this and after Campbell reason we can right here Len Peas 36 minus two x Keogh, where it is 26 minus two x and high days. As we can see in figure, it is very less so I can write Hiatus X and once again cushion be. We need to right domain for this function volume. So for that we need to We knew to cut something so that this ax is getter than zero. But we can&#x27;t. Kurt. More than half of the site. This is the condition given Haavisto, site of the scientist Lentil Metarie. We need to occur something on the But there is condition. We can&#x27;t got more than the half of the letter of the material. So we can also right exes less than 36.2 or ready. This is the moment. And in this third cushion that been See, we need to use graphing utility to draw our table drawer table. He was in graphing utility weekend draw or table below. So from graphing utility, we have this table here. We can see that the maximum maximum volume seems do arca between 52 So the five and seven Sunday we don&#x27;t here we can see by and seven centimeter. We have Mexico. There&#x27;s a new effects here. We have maximum volume and in the foot cushion that is in D. We need to use a graphing utility and we need to have a graph. So after using graphing utility, we have a graph for this. So this is the graph from caffeine utility here. We can see the volume is maximum about six. So we can ride this one. No maximum workers at exactly X is equal to six centimeter with walling began right here with the volume Ah, 33 the t 456 We can get here city for 56 centimeter cube.

Peter Winans · Answer

all right, if we have rectangle piece of material such that it is the dimensions Air six by a And if I am going to cut a tab out of each corner, that&#x27;s a square with dimensions of X by acts meaning we&#x27;re cutting out tabs that looks something like this. Then what&#x27;s gonna be left? And if we fold these tabs up after we cut these rectangles, we&#x27;re going to create a box. So the width of this box right here is gonna be the length six minus the length of each tab that we cut out. So it&#x27;s kind of to be six months to acts in this length here, from here to here is gonna have to be eight minus two acts. So in other words, what we&#x27;re going to get is a backs that will have the dimensions six minus two, x eight minus two x. And these tabs give us sides that are have a height of acts. So the volume of this box, I would have to be acts time. Six minutes to acts times eight minus two acts. And if we were to multiply that out, we&#x27;d have a nice cubic function that we would have to graph. So when we think about the domain, we can&#x27;t have XB longer than the sides of the material that we&#x27;re working with. And more specifically, if you think about Ray here, if we were to cut this side in half, then each piece would have to have a length of three. So we can&#x27;t have X be longer than three because it would be impossible to cut out two tabs, right? And in fact, even effects were three. We&#x27;d end up cutting this thing in half, and we wouldn&#x27;t have anything left to make a box out of. So our domain for this function, we&#x27;re gonna want to be the interval from 0 to 3. Okay, X is gonna need to be between zero and three in order for us to be able to make a bags. So if we were to graft dysfunction between zero and three, you&#x27;re gonna find it&#x27;s gonna look something kind of like this. And if we have our windows set up, such that we go up to at least 40 will be able to see the maximum point on this graph. So if you&#x27;re playing around with a graphing calculator Right now, you can find that maximum value, and it&#x27;s going to be at the coordinating 1.13 24.2 secs. So that means the length of the tab that gives us the greatest volume was 1.13 and the max volume that we&#x27;re gonna have is gonna be approximately 24.26 groups 24 0.26

Carson Merrill · Answer

so we know that volume is length times, width, times height. As that tells us that volume is going to first equal, it&#x27;s gonna have equal the the length, which is eight minus chu x times the width, which is 15 minus two X And then lastly, it&#x27;ll be multiplied times the height, which is just X. So that&#x27;s our volume function. Then we want to find the domain of E. We see that X is greater than or equal to, um zero. And it&#x27;s also, uh, it&#x27;s in between That&#x27;s and, um, 7.5 right here, because X cannot obviously equal a negative volume or anything like that. But it also has limits with, um, how large it can be and actually want to plot the graph of the function of V. Um, so here we have our graph that we&#x27;ve, um, modeled. And then, um, the domain of is actually me zero to infinity, cause as we see as we&#x27;re going up, it&#x27;s just gonna keep getting larger in terms of the X value. Um and then lastly, we want to describe how the volume varies with X, um, and describe how that someone could construct boxes of maximum volume. So we see right here that, um, you can maximize volume. Uh, if we make X equal to 1.667 we see that as X increases it&#x27;s going to here by multiplying times that it&#x27;s gonna increase the volume of the box. But we also have minus two X here. So there&#x27;s a certain point at which, um, if we zoom in right here, we see that the domain is restricted to four. In a sense, even though the graph is goes higher than that, this is really what we have to work with. Uh, because there is a limit to how much, Um, with how much material we have. So ultimately, this is what we&#x27;re focusing on. And it&#x27;s houses that we could construct a boxing with maximum volume if X was equal to 1.667

Daphne Pusey · Answer

in this problem. We have a piece of cardboard that started out as a basic 10 by 15. Okay, On one end, we cut squares out and the other end rectangles. What&#x27;s gonna happen is you&#x27;re gonna fold it up. Hey, you could see where I have marked base. You&#x27;re gonna fold things up and there&#x27;s gonna be a little on top, so it&#x27;s going to give you an enclosed box. All right, so the first thing we wanted bind is a formula for the volume off this box. OK, so we do know basic. Ah, volume formula. Okay. Is length times with times height. We know that. Okay, um but what we&#x27;re gonna dio is we want everything in terms off one other variable. So v is equal to something in terms of X. Let&#x27;s site. So what we want to dio is find a substitution for each one of these length and width and height in terms of X. So we&#x27;re just going to say, while we caught ex chunk here, we know this up. Here&#x27;s a square, so this acts by ex here, all right? And this is actually going to be our height, so that&#x27;s pretty easy height is just X cause this is the height of your box. Okay, while in terms, this is also going to be the height of our box. So will label that. Now we&#x27;ll think about that in a second. All right. What I want to look at is whipped, okay? And with is going to be this area from here to here of our box. Okay? What we have is w We started with 10 and we subtracted off two axes, so that&#x27;s not too bad. Now, as far as the length of our box goes, our length is going to be this here across where our bases. Okay, so that&#x27;s gonna be your length, But this is also going to be a length from here to here. So what we want to do is we&#x27;re going to say, Well, we start with 15. Okay, So 15 is equal to we have to exes. We haven&#x27;t X here. We have an X here, so we have two axes and then also plus two l&#x27;s. So when we saw that out for L. Because we wanna have l in terms of acts, we&#x27;re gonna end up with 15 minus two X divided by two. All right, so now, at this point, we can start substituting in to get an actual formula for volume. So length is 15 minus two x divided by two or with is 10 minus two X and our height is X. All right, so we&#x27;re going to do some basic algebra, and you&#x27;re gonna distribute that all out. What you end up with is volume is equal to two. Excellent. Third, minus 25 X squared plus 75 acts. Okay, so that&#x27;s the first part we had to find our formula. Now we need to find the domain. So to find the domain off this, um, formula here, what we want to do is actually ah, so for X basically So what we&#x27;re gonna do is plug zero and for V. And when you do that, you&#x27;re gonna end up with X being equal to zero five and 7.5. Okay, because we have a cubic function here. We need three answers. So we come back to our picture and think about this logically, Can x be zero? Well, not really. Because then we wouldn&#x27;t really have anything to fooled. Okay, so it can&#x27;t be zero, but it has to be X has to be greater than zero. So anything greater than zeros fly? Hey, can x be five if we come over here and plug in five. If we have a five here and a five here, five plus five is 10. That&#x27;s gonna be the same length or with as this cardboard originally started with. So X has to be actually less than five. Anything less than five. And we just said it has to be less than five. Therefore, 7.5 in this instance is not a violent answer. All right, so what we want to do, then, is we kind of wanna make a quick sketch or graph Lis. Hi. And we&#x27;re gonna have 2040 60. And you would take go online and look up a graphing site. Or if you have a handheld graphing calculator. Okay, we know we&#x27;re gonna hit zero at five. Okay? So when you graph this, you&#x27;re gonna have a line, comes up just above 60 a little bit comes back down, it comes back down, But after the five, we really don&#x27;t care what it does because that&#x27;s not in or domain All right. Well, we want to do is find out what that point ISS. You go ahead and we&#x27;ll cut your graphing calculator. You&#x27;re gonna find that that point is 1.96 and 66.2 Now, just to confirm this, we&#x27;re going to take the derivative of this here. So the derivative is gonna be six x squared, minus 50 X plus 75. All right, we&#x27;ll solve this for X, and I would use the quadratic formula. That&#x27;s probably gonna be the easiest one to use quadratic formula. And you&#x27;re gonna find the axe is going to be equal to 6.37 and 1.96 Okay, 1.96 we had here. Yes, confirmed. But it also gives us 6.37 What? Our domain is from 0 to 5. Therefore, that answer is too high and invalid. Um, so are X is 1.96

Problem

A roofing contractor is fabricating gutters …

Problem 98 Easy Difficulty

Answer

a) $8 x^{3}-144 x^{2}+576 x$
b) $0 < x <6$
c) See graph
2.5

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a) $8 x^{3}-144 x^{2}+576 x$b) $0 < x <6$c) See graph2.5

Precalculus with Limits

Grace H.

Catherine R.

Alayna H.

Michael J.

Absolute Value - Example 1

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Ron LarsonPrecalculus with Limits

Grace H.

Catherine R.

Alayna H.

Michael J.

a) $8 x^{3}-144 x^{2}+576 x$
b) $0 < x <6$
c) See graph
2.5

Ron Larson

Precalculus with Limits