in-the-following-exercises-solve-the-given-maximum-and-minimum-problems-us-postal-service-regulation

Question

In the following exercises, solve the given maximum and minimum problems.
U.S. Postal Service regulations require that the length plus the girth (distance around) of a package not exceed 108 in. What are the dimensions of the largest (volume) rectangular box with square ends that can be mailed?

Catt Huth · Accepted Answer

I have a lot going on this problem. So the first thing we need is a box with a square in despots. Six seconds Already been through the postal Service cleaning appear a little bit, all right. And they&#x27;re telling us that we don&#x27;t know the length of this. A pair of, like, measurement here. It tells us the girth. That&#x27;s this area around it. So girth in actuality is pretty much the same thing as perimeter. All right, so they tell us this girth at mass could be 100 and eight inches. So 108 inches is less than using the idea of perimeter to s plus two. They tell us this ass is 18 inches. Our side is 18 inches. So it is so for else that multiply that by two, subtracted it from both sides two times 18 36. So subtracting that I get 72 is less than or equals that too, too. Well, divide both sides by two and 36 is less than or equal to. So we really could have done the spine equation. But we&#x27;re leading into our inequalities here a little bit, but now they want to know is what&#x27;s the maximum volume that I could get? All. So we&#x27;re just going to use our volume equation and volumes equal to the base times height. Remember this basis, that area of the base. That&#x27;s why it&#x27;s a Capital B instead of a little B. Yeah, we could replace that h within out. You&#x27;re right and already a substitute in. So we know that our base, if I may use that square, is 18 inches so that 18 inches squared and I&#x27;m gonna multiply that by my length, which I found it in a maximum of 36. Do the math like 11,664. Remember units inches cubed. And so our maximum value is 11,664 inches. Cute if we&#x27;re using a box with a square base measuring 18 inches, Other measurements. Mike is awesome.

Timothy Stampf · Answer

were asked to find the length and mask maximum volume of a package that could be sent through the US priority mail system. Now we&#x27;re already told that one end of the package, which is in the shape of a rectangular prism, is 18 inches by 18 inches, as shown on this diagram. And the girth, which is a terminology specific to the post office and shipping companies, is defined as Thebes distance around the end of the package. So that sounds very similar to Perimeter that we&#x27;re familiar with. So the girth for this package would be 18 18 18 and 18 this four times 18 which is 72. So by definition, the girth on this package is 72 inches. Now the post office has established a rule stating that the length plus the girth of any package, cannot be more than 108 inches max. So I guess we could also say has to be less than or equal to 108 inches max. Either either way will work for this problem. You want to use less than or equal to or equal till that&#x27;s fine. All right. Now, the first quantity we&#x27;re gonna calculate is the length once we have the length than we can calculate the volume. So we&#x27;re gonna use thea relationship that the post office uses to restrict the size of the package to calculate what the length ISS. So if the length plus the girth equals 108 we&#x27;re gonna use the equal sign because we&#x27;re going to assume that this is the maximum size package. And that&#x27;s what the problem is asking us for the maximum size so maximum is equal to 108 now, to calculate the length, we simply subtract g from both sides of the equation, and we can calculate the length is being 100 and eight minus 72 and the length turns out to be 36 inches, and that&#x27;s Max. So a package that&#x27;s 18 inches by 18 inches by 36 inches is the maximum size that we can send via US priority mail. And the last thing we need to calculate is the volume of this maximum size package. And that volume, or the formula for the volume of a rectangular prison, is the area of the base times, the haIf So we look at our rectangular prism will consider the ends that are 18 by 18 that we used to calculate the girth. We&#x27;ll consider that our base. So our base would be 18 inches by 18 inches, and then the heists would be the distance between, um, the two bases. And in this case, that would be what we have labeled L on the diagram, and that is 36 inches. Now we calculate all of those numbers. Together we get 11,000 664 inches cubed, and that is the maximum volume of a package that we can send through the U. S. Postal says system with a square base in our rectangular prism of 18 inches by 18 inches.

Sherrie Fenner · Answer

every question. 29. We have a rectangular box and our and is a square. So our volume is gonna be X squared times why I cannot draw three dimensional. Um, and we know that our is 100. Just the perimeter is equal. Uh, why plus two x plus two x so 108 inches. Or we could say why he won&#x27;t 108 lines for X. And now I&#x27;m gonna put this 108 minutes for X back into my volume. So we is going to eat more. Thanks were times a minus X which gives me on and the Eaton&#x27;s. I want a X weird minus four x cute. And I want to take the derivative of that to find the organization for Have the room, the Coons. 216. Thanks, minus 12 x there. And now we just solve it like we would go and, um, Rambla or any quadratic my I guess the easiest. I think I&#x27;m just gonna write it over here isn&#x27;t gonna factor out in X shame on the factor. Yeah, just thanks. So either X equals zero or to 16 minus 12. X equals zero. So negative. 12 checks Negative too. 16 by both. I&#x27;ve been negative. 12 therefore expertise for 18 inches. When you plug that back into your y equals you get why equals one? 08 minus four times 18. Yeah, for why peoples?

Regina Hays · Answer

Okay, Um, the U S post office, um, allows any boxes sent through the mail to have a length plus what is called a girth totaling no more than 100 and eight inches, so l plus G has got to equal +10 wait. Now the girth. Okay, the girth is actually the combination of the height and the width. So this is with and this is a TSH um, G is actually equal to the perimeter, which is to h plus two w came. And so we want to find the dimensions of the box. So we want to determine the dimensions of the box with maximum volume. Okay, um, and the maximum volume is also we&#x27;re gonna let w equals h, so we&#x27;ll let that be a square girth. Okay, So since we or maximum maximizing volume, we need to come up with the volume equation. And so volume is equal to W. Times, Age times L which w. And H. They&#x27;re the same. So we can just let it be, um w squared l You could have used eight square del if you wanted to. A swell and for that matter, we can go back and say girth. Since H and W is the same, this is really four w Okay. And so what we&#x27;re going to do now is to, um we want to take the derivative of the volume equation. Um, boat. Before we do that, we want to only have it depended upon one variable. So this is where we&#x27;re going to have to come up with another equation that we can actually solve for that variable would. And so I noticed that ALCs actually have appear We actually have l plus four w is equal to +108 And so we can say that link is equal Toe 108 minus four w. And so I can use that in here to change my volume formula only in terms of the with and so w squared is times 108 minus four W. And I know that my with has got to be bigger than zero. Okay, so let&#x27;s go ahead and start with that. So I have a volume formula that is only dependent on W and that waas and, um, next time I&#x27;m a writer, I&#x27;m I would go ahead and distribute that w squared show it becomes, Ah, 108 W squared minus four w cubed. So there is my volume formula. Now I&#x27;m going to take the derivative with respect to W. And that becomes to 16 w minus 12 w squared. Okay, so now we want to find the critical values. And we do that by setting the derivative equal to zero and solve for W. And so I&#x27;m gonna factor out a 12 w. And that gives me, um, 18 minus w equal to zero. So that tells me w can either be zero, which, um, we know that it can&#x27;t be or w can equal 18. And so this is telling me, and especially since my domain is open ended This is telling me that my maximum volume is gonna occur when my with is 18 inches, which now means my height is 18 inches because they&#x27;re the same. And now my link is 100 and eight minus, um, minus, uh, forgot What? Itwas now It was 100 and eight minus four times 18 which gives me 36 inches. So a box will be 18 by 18 by 36 inches to maximize volume

Carrie Bain · Answer

I couldn&#x27;t draw the diagram of about C here to help us think about this problem. Let me draw that better. Let&#x27;s joining the courtesies here. Those don&#x27;t look like they&#x27;re joined. Very well. Let&#x27;s see. Going to join the corresponding Bartosz ease to former a rectangular prism. Here we go. Now this is the link. This is the were And this is the heart. Now the length plus the girl can be no more than one away. So your link plus the girth Now the girth iss the perimeter around this back So the width and the height So what you&#x27;ve got is to dub yah plus two terms. The height equals one away. Now you were told in the problem that the width and the Haider equals So what we actually have here I&#x27;m gonna let it&#x27;s be the were And it just so happens that is the same as the heart. So what we have here is we have the length plus now this to debut in two age, it&#x27;s like two x plus two it so that&#x27;s a for its equals. One of eight. Now, when we saw firmly in terms of eggs, that will be 100 and eight minus for were it and we want to maximize our volume and volume is length tons with tons height. So our volume in terms of eights IHS 108 minus forward. It&#x27;s Tom&#x27;s, it&#x27;s and then times and other ends. So this is 100 in a minus forward rates. That quantity is times it&#x27;s squared. So our volume in terms of eggs, it&#x27;s 100 in eight ants squared, minus for war. It&#x27;s Keat. Now we want to maximize the volume, so I&#x27;m gonna take the derivative of that volume that will Bay 200 in 16 eights Manners. 12. It&#x27;s squared, and we&#x27;re gonna set that derivative equal to zero. Now there&#x27;s a common factor of four. It&#x27;s that I can pull out and to sit stained about it back for war. It&#x27;s 50 forward and then 12. About about forwards. 30. We got an immense left there. Now this means that you&#x27;re for its ankles, a robe or 54 monness. Three. It&#x27;s ankle zebra. Well, for its equals, the right. When you divide by for work, you get the dates equals zero, and that&#x27;s not know domain. If it&#x27;s worthy road, we wouldn&#x27;t have a width throughout. Here we have 54 minus three X What&#x27;s at three ext. About side. So what you have is 54 equals three h. Divide by three, and you&#x27;ll get that X is 18. Now we have what we need to find the three dimensions. Remember that adds, was your wit and your hot. That&#x27;s what that was the variable we used to represent. But the way it and the height. So the width and height are both 18 inches. Now let&#x27;s use the expression we found for the lanes to give that number and are linked. Waas ah, 108 minus four It&#x27;s and so that&#x27;s 108 minus four times 18. When we do that calculation, we get 36 inches, so the dimensions are

Problem

In the following exercises, solve the given maxim…

Problem 24 Easy Difficulty

In the following exercises, solve the given maximum and minimum problems.
U.S. Postal Service regulations require that the length plus the girth (distance around) of a package not exceed 108 in. What are the dimensions of the largest (volume) rectangular box with square ends that can be mailed?

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Anna Marie V.

Heather Z.

Kristen K.

Precalculus Review - Intro

Functions on the Real Line - Overview

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Basic Technical Mathematics with Calculus

Grace H.

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Grace H.

Anna Marie V.

Heather Z.

Kristen K.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus