![]() ![]() 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. 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. If you liked my explanation, I’ll make more. 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 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. 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. Form this box in particular using a sheet of printer paper. Convert the values to centimeters since they’ll be easier to measure out. The reason why I chose this particular situation is because you can easily see this yourself with an actual sheet of paper. Which with a little calculator work is easily seen to be about 66.14824.įrom 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. We simply plug in our value of x into our original volume function. So to maximize the volume of our box, we’d need to cut out squares of side length 1.58542 inches from each corner.įinding the volume of the box is a very obvious and very simple process. Thus, we know that our optimal value of x is not 4.91458. We can’t snip away 9.82916 inches away from 8.5 inches for obvious reasons. Our paper’s shortest side length is 8.5 inches. We would have to snip away one of these two values: That means, from each side, we would have to snip away twice the critical value. From each side, we’re snipping away 2x, as can be easily seen if one draws a picture of this. I assume you know your basic algebra, so clearly this’ll outputīut remember. Because this is clearly not a factorable polynomial, you must use the quadratic formula. To find the critical values, we must first derive V(x).Īnd then set this equal to 0 and solve for x. Now we have to find the critical numbers for the volume, because that is where the values of x can be optimized. To make this easier to derive, we distribute to turn it into a cubic polynomial (expanded form). ![]() Obviously, the volume of this box will be: So what is this box’s volume? The volume of a box is obviously its length times width times height. If you have trouble with this, I can do so for you. The height is simply x, as can be seen if you actually make the box, or draw the picture out. That means the side lengths of our box will be (8.5-2x) and (11-2x). We know that from each side, we’re removing 2x inches (because we’re removing from each corner). ![]() We know a standard sheet of paper is 8.5 by 11 inches. Now what value of x will give us the maximum volume, and what is the maximum volume? (If this is hard to imagine, take a sheet of paper, cut squares out from the corners, and tape the adjacent sides together). ![]() 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. Imagine you want to create a box from a single sheet of printer paper that maximizes the volume you can get from it. Really, there’s a higher chance that any error you make is algebraic rather than in the calculus itself. For now, I’ll just post my explanation about optimization problems here, for anyone that needs help in them.Īs said to him, they’re awfully easy once you get the hang of them. But a rant about how much I hate Common Core is for another post. The rest of this Common Core stuff is just Complete Crap. 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. He might have been a few days ahead of me in the curriculum. Turns out a few days later, that’s what was taught in my class. At the time, I didn’t know how to do them. Someone asked me to help them with optimization problems earlier. It’s not because I forget what to do… I usually run out of time because I’m very OCD about the details. Although, as much as I understand these types of problems, infallibly I do poorly on the tests. 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. ![]()
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