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A box with square base has an open top and a volume of 200 cm3_ The base dimension, X, is a variable measured in cm, The height is also variable, but constrained by...

Question

A box with square base has an open top and a volume of 200 cm3_ The base dimension, X, is a variable measured in cm, The height is also variable, but constrained by the total volume, as shown in the figure below:200The surface area of the box (base and four sides) is a function of X: 200 A(x) x2 + 4x1. Graph the function A(x): Assume xHow many critical points does have?What value of x gives the smallest possible surface area? Be accurate to three decimal places_What is the minimum surface area?

A box with square base has an open top and a volume of 200 cm3_ The base dimension, X, is a variable measured in cm, The height is also variable, but constrained by the total volume, as shown in the figure below: 200 The surface area of the box (base and four sides) is a function of X: 200 A(x) x2 + 4x 1. Graph the function A(x): Assume x How many critical points does have? What value of x gives the smallest possible surface area? Be accurate to three decimal places_ What is the minimum surface area? Be accurate to three decimal places Amin Dimensionally incorrect: Please check the type or dimension of your unit: Enter a number with units



Answers

Minimum-surface-area box All boxes with a square base and a volume of $50 \mathrm{ft}^{3}$ have a surface area given by $S(x)=2 x^{2}+\frac{200}{x}$ where $x$ is the length of the sides of the base. Find the absolute minimum of the surface area function on the interval $(0, \infty)$ What are the dimensions of the box with minimum surface area?

Okay, so we want to find our minimum surface, so we'll stop taking our derivative and finding are critical voice. So we get four X plus native 200 x A gram negative too. Okay, let's hear it. This is for X plus. I should. One is 200 over X squared. Now let's multiply this by X squared over a squared. So this is equal to four X cubed minus 200 over X squared. And that set the seafood job now, can we factor are numerator. Ah, that's for Exeter three minus 200. Okay, so we see that we can factor out of four and get X cubed minus 50. So we want one. X cubed minus 50 is equal to zero. So X is equal to Cuba of 50. Okay, now let's begin. This ex points defines our ah minimum surface area. So s square roots or a key route of 50. That's equal to two times he route. If it b squared minus plus 200 over the cube root of 50. And now plug this into my calculator. Okay? And I gives us our minimum surface area is 81 points for three

For this problem we are told that all boxes with a square base in a volume of 50 cubic feet have a surface area given by sfx equals two X squared plus 200 over X. Where x is the length of the sides of the base. We are then asked to find the absolute minimum of the surface area function on the interval zero to infinity. And we are asked what are the dimensions of the box with minimum surface area. So our first step is to take the derivative of S with respect to X, Which is going to give us four x minus 200 over x squared. We want to solve for one that equals zero. So first step here is we can multiply through everything by X squared giving us four times X cube minus 200 equals zero. Or we can then divide everything by four. So we would get x cubed minus 50 equals zero. So we need that X cubed equals 50. So X is going to be the third root of 50 which comes out to let's see here, that is approximately 3.684. Now we're told that we have a square base and one um the or rather we have a square base and X is the length of the sides of the base. Where that would mean that our volume is going to be x squared times our height or whatever we want to call it. But X squared times height needs to equal 50. So H would be equal to 50 divided by x power of 2/3. But That means that it will be 50 divided by 52, the power of to over three, which means that H must equal X And in turn must equal 50 to the powerful over three.

A piece of cardboard is twice as long as it is wide And it is to be made into a box with an open top by cutting two in squares from each corner and folding up the sides. We're going to let X represent the width in inches of the original piece of cardboard. The first thing we're gonna do here is represent the length of the original piece of cardboard in terms of X. So we are going to let X be the with which then means because it was twice as long, two X is the length. Next we're going to figure out what the dimensions of the bottom rectangular base of the box will be. We're going to give these restrictions with X. So when we cut out the 2" squares which are represented right here, these pieces here. So you cut off those 2" squares and fold up the sides. The dimensions of the bottom of the box is X -4. And the reason is two plus two is four. So we have to take that four hours. So we get x minus four And the length is going to be two X -4. We're taking two out on this side, two plus two is four. So would be x minus four. We're taking two plus two out on this side and we said this is two X. So it becomes two x minus four. And so the restrictions then Are that it's between infinity and four. Yeah. Next we're going to look at and determine the volume that represents the volume of the box in terms of X. So volume is lunch times with times height. So we get volume is the X -4 Times two x -4 Times to those 2" squares we cut out and now we're going to expand this out. We get X times to access to X square negative four times two X is negative eight X X times negative four negative four X a negative four times negative four is 16. And all of that times two I'm going to clean this up by combining the like terms. 1st we get two X squared minus 12 X plus 16 times two and now we're going to distribute that out and we get four X squared minus 24 X plus 32. And this is the function that represents the volume of the box. Next we're gonna figure out for what dimensions of the bottom of the box Will the volume B 320 inches. So we get 320 equals that function for the volume four X squared minus 24 X plus 32. all of these numbers are divisible by four. So I'm going to Divide Everything by four. Just to make it easier to work with and we get 80 equals X squared minus six X plus eight And thus attract 80 from both sides. This gives us zero equals x squared minus six X minus 72. And now we're going to factor this and we get X -12 Times X-plus six. And this would mean that X -12 equals zero or x plus six equals zero At 12 to both sides. We get x equals 12. Subtract six from both sides And we get x equals six. It's going to be the x equals 12. It cannot be x equals negative six because it's negative. It's excluded from the restrictions that we had right here cannot be a negative number. So it can't be this one. This is excluded from the domain. And lastly We're going to find the volume of X such that the box has a volume between 400 and 500. So to do this we actually have to look at the 400 And the 500 separately. We're going to do the same thing like we did here Where we put the 400 or 500 right there instead of 320. And we're going to get 400 Equals four x square -24 x plus 32. and for the 500 we would get 500 equals four, X squared minus 24 X plus 32. Let's look at the 401st. Everything divided by four. So let's do that. We get 100 equals X squared minus six X plus eight. Subtract both sides by 100 and I get zero equals x squared minus six X minus 72 minus 92. That should be a nice. Alright, so zero equals x squared minus six. X minus 92. And now we can use the quadratic equation X equals negative B cluster minus the square root of B squared -4 A. C. All of that divided by two a. This gives me x equals six plus or minus 36. And if we take negative six squared minus four times one times negative 92 We actually get 404. All of that divided by two times one is just too. So x equals six plus 20.01. The square to 404 is 20.01 over two or X equals 6 -20.01/2. So x equals 13.05 or X equals negative 7.5. Well it can't be this one. It can't be negative excluded from the domain. So we're left with x equals 13.5. Going over here to the 500, I'm gonna divide everything by four. Again, clean it up. It makes it easier to work with And we get 125 equals x squared minus six. X plus eight. Subtract 125 from both sides. We had zero equals x squared minus six x minus 117. We now have a quadratic formula. We can use the quadratic equation X equals negative B. Foster minus the square root of B squared -4 times a times c. All of that divided by two a X equals six cluster minus the square root of 504 over to. So x equals six plus 22.45. The square root of four, is 22.45. Approximately over two or X equals 6 -22.45/2. So x equals 14.23 Or x equals negative 8.25. It can't be negative 8.25. That's excluded from the domain. So we're gonna rule it out and we are left with the X equals 14.23. So the interval on which it could have a volume between 400 and 500 Is between 13.05 And 14.23. And this is an inches

Okay, So, um, this question we have an open box is to be constructed from a piece of cardboard 20 centimeters by 40 centimeters by cutting choirs of sight of X from each corner and folding up the sides as shown in figure a, um, express the Volume VII of the box as a function of X. So that's expressed the volume of the off the box as a function off ex. Okay, so I'd draw a graph of the box here. Um, you see, I've already put on on some of the lens. I don't just explain where I got these from, so of course you got our box was 20 centimeters by 40 centimeters. Who don't know why? That's just atlas that strange. I'm just undo that real quick. They also got 20 centimeters by 40 centimeters. Um, and then we had corners cut out of length and width. Ex. These are colonists. They would let them with X, and then I've got this 20 minutes to accident. It's 40 minutes to exit the way I got these because I considered the fallen thing considered how much of the length has been cooked to get these sites so the full length is 40 and we cooked to excess of this length here on this lens on highlighting here. This is 40 minus two X in this width is 20 minus two X. Um, and find the volume. If you imagine folding this box together, our height is gonna be extends. Its volume appears just gonna be a high time. That with time, the value I wit, um imagine folding it opens. It's gonna be 20 minus two x and then, ah, um, length is every 40. Most jerks. I know a lot of people don't actually have great sort of, uh, imagination when it comes to. So I've actually how this looks is a box. These lines may help you imagine. Start folding these upwards. Um, you could even in a tactical, even start like, Ford piece of tracing paper or something, just sort of try and see what it looks like. Yeah, it can be quite difficult, Isn't it? Unfair? Sometimes if you can't see it, then you'll just have to use all the various methods to try and see it. There we go. The volume is gonna be X times by 20 minutes to extract by 40 minus two. So that's how you in terms of X and there we go. That's the answer by a B. It's actually use to find a domain for on V. So what is the values that excuse? First off, I'm just gonna draw a little graph of what V would look like. Okay, Um all right. So we're gonna have one turning 10.0 here because of the ex. Could have another one at 10. Because of 20 minutes to X, we'll have another one at 20. This is going to be a positive X cubed graph. So this is gonna look a little something like this. The court from the bottom here. Not a turning point. I don't even know the Chinese point. I'm gonna exercise, uh, intercepts it. It's why graph looks like asking for I don't mean so that's the range of what the exes could be. You can't have anything past here because that would be a negative one thing. You can't have a leg. I'm just emphasizing here that we have on X on our except system. We have V on our wise. That's volume on. Let's all hyped in this case, um, and Also, we can actually have a height greater than 10. Because if X is greater than 10 then we're gonna get a wit. That is zero. Because you imagine our, which was 20 minutes to act so that X is greater than 10. We're gonna get whipped of zero, and that is not feasible. So it cannot be passed here either. So I domain is actually gonna be between zero and 10. And you can't have a length of zero. It has to be at least greater than zero. So it's not even equal to so I domain is actually going to pay. Um, X is greater than zero on less than not even equal to extraordinary. Greater than Aunt Lester. There goes. That's part B. And then finally, Patsy's actually draw a graph the function of it and use it to estimate the maximum value for such a big box. OK, body count's graph able, Just draw again on just be extra clear. 10. 20 on that. The origin joined. I got there, uh, can't. Right. Um, So what I'm gonna do here is its estimate. Um, I'm looking at the question Do you think actually wants us to find crying accurate? Yeah, because you can easily say, Oh, it's just somewhere between 10 and zero. So it'll be five. SE x is five. It could be anywhere between, So I'm actually work without fully amusing RV. So first going to expand its out. So we're gonna get 20 times life or two here. First off that she's my calculator. 100 and a three and x because of the time fly the X So we just explained in these brackets out there looking at minus 80 x minus 40 ex SAS doing minus 120 x squared for this time all by X And then for X cube component, we're gonna get on plus four X squared tight line of the excellence plus for execute. All right, uh, we're gonna differentiate. It's not something that Devi vine t x really It well, that squares here minus off 240 x here and plus 100. Well, im just differentiating here if you the video's online to such a um how's differentiate, but just really quickly, I'm just times in by the power and the power. That's not enough time to set that equal to zero So you've already It's let equal zero by letting that equal to zero. I'm gonna find the ex corner of are turning points. Okay, equals zero. Just use my calculator hits. Sell this. If you have a calculator than the quadratic formula will work well, you might be able to fact arises. Well, maybe dividing by 12. And that might give you an answer. Although, uh, you get quite dodgy. Answer. Yes, I'd imagine I'm probably factory ized to hear we got X coordinates is either 15.77 Which two? Large, actually. So we can cross this one out. We got another one as four point. Ah, 22 six on something. So, uh, this will be our one, cause that's actually within that domain on. So this will be the one that we're using 4.2 to 6 on the question asking the maximum volume for such a box. We saw these back into here. Um, because you got all we need. That X I we don't need to find. Yeah, I'm to find out the we're gonna solve it back in. Yeah, So I'm gonna sew back in. So that's 4.262 to 6 by 20 minus two times by 4.2 to 6. And I'm just gonna put this into my calculate your hair. I've already got the answer. So if you want to check it, you put a calculator as well. I trust you over me. Honestly, Um, there we go on, and that will give you an answer of 1539 0.6. A warning that three centimeters. Cute. You know, that's the answer. If you have any questions about that, you could even be a donor. Dr. J. P H I don't know. At Gmail. Don't come. Thank you very much for listening. And I wish you the best facilities.


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