Optimization Problems [Paper Box Folding]

As ExperienceProject’s math tutor, if it’s Algebra 1 and beyond, and if I’m able to understand it, or if it’s in my capability to learn what’s being asked of me, I shall teach it. Although, as much as I understand these types of problems, infallibly I do poorly on the tests. It’s not because I forget what to do… I usually run out of time because I’m very OCD about the details.

Someone asked me to help them with optimization problems earlier. At the time, I didn’t know how to do them. Turns out a few days later, that’s what was taught in my class. I’m assuming he’s in AP Calc AB too. He might have been a few days ahead of me in the curriculum. I guess the only good thing about Common Core is that everyone’s on the same page… literally, in a way, give or take a few days. The rest of this Common Core stuff is just Complete Crap. But a rant about how much I hate Common Core is for another post. For now, I’ll just post my explanation about optimization problems here, for anyone that needs help in them.

As said to him, they’re awfully easy once you get the hang of them. Really, there’s a higher chance that any error you make is algebraic rather than in the calculus itself.

Here’s how to do optimization problems:

Imagine you want to create a box from a single sheet of printer paper that maximizes the volume you can get from it. You create a box by cutting out squares with side lengths x from the corners, then fold the little tabs formed upwards, and taping the box together. (If this is hard to imagine, take a sheet of paper, cut squares out from the corners, and tape the adjacent sides together).

Now what value of x will give us the maximum volume, and what is the maximum volume?

We know a standard sheet of paper is 8.5 by 11 inches. We know that from each side, we’re removing 2x inches (because we’re removing from each corner). That means the side lengths of our box will be (8.5-2x) and (11-2x). The height is simply x, as can be seen if you actually make the box, or draw the picture out. If you have trouble with this, I can do so for you.

So what is this box’s volume? The volume of a box is obviously its length times width times height. Obviously, the volume of this box will be:


To make this easier to derive, we distribute to turn it into a cubic polynomial (expanded form).


Now we have to find the critical numbers for the volume, because that is where the values of x can be optimized. To find the critical values, we must first derive V(x).


and then set this equal to 0 and solve for x. Because this is clearly not a factorable polynomial, you must use the quadratic formula. I assume you know your basic algebra, so clearly this’ll output



But remember. From each side, we’re snipping away 2x, as can be easily seen if one draws a picture of this. That means, from each side, we would have to snip away twice the critical value. We would have to snip away one of these two values:

2x≈3.17084 inches.

2x≈9.82916 inches.

Now one needs to think logically. Our paper’s shortest side length is 8.5 inches. We can’t snip away 9.82916 inches away from 8.5 inches for obvious reasons. Thus, we know that our optimal value of x is not 4.91458. Our only value of x left is 1.58542. So to maximize the volume of our box, we’d need to cut out squares of side length 1.58542 inches from each corner.

Finding the volume of the box is a very obvious and very simple process. We simply plug in our value of x into our original volume function.



Which with a little calculator work is easily seen to be about 66.14824.

So our final answer would be:

From a standard sheet of printer paper that’s 8.5 by 11 inches, we can create a box with a volume of about 66.15 cubic inches by cutting out squares from the corners with a side length of about 1.59 inches in order to create an optimal box where we can get the greatest volume possible from a paper of these dimensions.

The reason why I chose this particular situation is because you can easily see this yourself with an actual sheet of paper. Convert the values to centimeters since they’ll be easier to measure out. Form this box in particular using a sheet of printer paper. Then try and make other boxes by cutting out different values of x (smaller and larger squares) from the corners, and see the difference in volume.

If you don’t want to waste paper, all you’d need to do is pick a value for x that it is possible snip out of the paper (in essence, a value between 0 and 8.5), and find the volume of the box for that value of x with the volume function. You’ll see that none of the other values will allow for a volume greater than or equal to the critical value in our range.

If you liked my explanation, I’ll make more. Just mention the topic, and if it’s within my capability to learn it or explain it, it would truly be my pleasure to do so. Seriously. For some odd reason, I got suspended from EP for a few days (that site has several glitches), perhaps due to someone feeling offended by a 3 year old post of mine and flagging it. Of course, an email to the moderators quickly restored my account. The mods are aware of my helpfulness as the community’s math tutor. In the five days without the site, I felt very uncomfortable and bored. It’s in my nature to teach math. I just have to. I love to do so! I dream of eventually becoming a professor of mathematics at MIT one day!  I seriously don’t know what it is about that particular college. I just love it so much! Maybe it’s because it’s where most of the cool technologies come out of, and my earliest screen name was “TheTechnoGirl.” Anyways.

In fact, I could type up what would effectively be the entire course of AP Calculus AB… you know, as I progress in the course. Typing up explanations ensures that I understand what I’m doing… it forces me to study. Oh hey! I just devised a studying plan! Every day a lesson is taught, I’ll come home and create an explanation to post. In explaining it, I’ll be forced to look at my notes to see if I’m doing it right, and to explain it properly. In fact, that’s what I needed to do for this optimization problem here and there.

And I’m not plagiarizing problems, either. I made this problem up myself, based upon a similar one given in class. Forcing myself to create the problems as well as solving and explaining them myself is more than hitting two birds with one stone. It’s more like I’m throwing a hand grenade into a pond of ducks (I remind you that I’m a bit GLaDOSesque–I have an awfully morbid and sardonic sense of humor). Not only will I be forcing myself to make sure I understand the material, therefore acting as my study session, whereas no other method of studying has ever worked for me before, I’d also be providing learning material for others that need it, along with satiating my need for teaching.

No one really understands my life. But really, no one but me needs to.

Optimization Problems [Paper Box Folding]

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