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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.

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

Algebra

Chapter 2

Polynomial and Rational Functions

Section 2

Polynomial Functions of Higher Degree

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

Missouri State University

McMaster University

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

04:40

An open box with locking t…

05:32

An open box is to be made …

03:40

(a) An open-top box is to …

03:03

As shown in the accompanyi…

06:53

Designing a box with a lid…

08:45

Volume of a Box An open bo…

01:58

An open box of maximum vol…

04:44

Constructing an Open Box A…

10:23

A piece of cardboard measu…

04:29

An open box with a square …

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't seen this before. That there, they're going to have thes corners. Traditionally, we just deal with these corners being axed, but they show on there is how they're actually gonna fold over the tab so that they make the box dirtier. So there, actually, the box they haven't shown like this. That and I'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'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'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'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't want to be in. We get 5, 576 and then we'll have right here. We'll have minus 48 X and then right here we'll have 24 times four. That's giving us minus 96 X and then multiplying. Here we get plus eight x squared and now we'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'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'm going to do a table of values now. What kind of values can we have? Well, let's look at this box. We can see that X naturally has to be bigger than zero. Otherwise, we'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'm gonna start my table at zero, and I'm going to go up by ones. And when I do, I'm just going to kind of look through the values and I can see that the highest looks like it's at three inches. At three inches, it's all the way up to 648. So I'm gonna set my graft to graft this. I'm going to go from 0 to 6. My ex access and my window is going to be from 0 to 6 and I'll scale by ones. And then I'm going to go up to I'll go up to 700 on this axis, so I'll have my wine minimum B zero. I'll go up to 700 scale by hundreds, and when I dio, I get this graph. It goes up and it's a little bit skewed and it comes down and I wanna find that maximum. So we're gonna do our second calculate and we're gonna find this maximum point. And it happens to be four on my calculator. I'm going to get to the left of it and enter. And then I'm going to get to the right of the maximum and enter, and then I'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?

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In mathematics, the absolute value or modulus |x| of a real number x is its …

An open box with locking tabs is to be made from a square piece of material …

An open box is to be made from a square piece of material 36 centimeters on …

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