5

7ptsQuesticn 6square basc. Thc sides 3r0 madc of plastic which costs You are building rectangular box with foot The metallic base ad top cost S10 per square foot: Y...

Question

7ptsQuesticn 6square basc. Thc sides 3r0 madc of plastic which costs You are building rectangular box with foot The metallic base ad top cost S10 per square foot: You have 5240 to spend: per squarc What is the maximal volume of thc box?200160Cujestonpts

7pts Questicn 6 square basc. Thc sides 3r0 madc of plastic which costs You are building rectangular box with foot The metallic base ad top cost S10 per square foot: You have 5240 to spend: per squarc What is the maximal volume of thc box? 200 160 Cujeston pts



Answers

A closed box in the form of a rectangular parallelepiped with a square base is to have a given volume. If the material used in the bottom costs $20 \%$ more per square inch than the material in the sides, and the material in the top costs $50 \%$ more per square inch than that of the sides, find the most economical proportions for the box.

So the topic for this video is to use the Grandjean multipliers to find maximum minimums. And here we are asked to find the max volume of a box given the surface area constraint or rather the cost of surface area constraint. And here the first step is to first draw a little picture of this box. I label the base X. Y. And the height Z. And we are given that the base cost 0.6 cents per square foot. But since the base is also given that it is three times as much to make, then sides or the top, Therefore the site and the top their cost base is 0.2 cents. 0.2 volume is X. Y. Z. For a box in a surface area. Well, Since the Cube there is two x wise, There are two wise ease And there are two. x.s. And therefore this is the surface area. But we don't want surface area, we want cost. And therefore what we simply do is simply multiply these sites by the cost to get the total cost. Since the bottom is the only one that is 0.6, I'll split this two X. Y. Into one X Y plus one X Y 0.6 XY. That covers the bottom 0.2 XY. That covers the top 0.2 x. z. That covers this, The front and the back. But since there's two multiplied by two And why is he covers the sides? But since there to multiply again by two and this should equal a grand total of 12. Therefore, if you subtract 12 this see is your constraint on this V It's your optimize function. Then all you need to do after this is to take the lead Groningen uh method by having the gradients of the equal to lambda, the gradient of C. So here, well, since lambda is simply just a constant, I can move, I can have the constant be either on the right side or the left side, I choose the left side. Just need to be consistent as you do with the problems. So what you get is when you do the partial derivatives, you get these three equations set an equal to each other and then use these equations to find The values to plug in four B. The trick to solving these equations as you stick is you choose any variable and then use that variable to express all the other variables here, I'll choose X and they're so first X is equal to X. That's simple now, so next, when you try to solve for one of the variables here that we will solve for why and to do that. The trick here is to use the system of equations to cancel out one of the variables, or at least one variable at one side of the equation. We're gonna use equations one and 2. And using equation one. And subtracting equation too. What you get is this and then subtracting Y minus X. And factoring what you get is the solutions. And therefore this implies that Y equals X. Or 0.8 equals Y. Z. Now only one of these works. And to show why this 0.8 equal spicy doesn't work If we substitute gamma land, that land dizzy. It was some suit land dizzy to zero And 0.82 equation too. What I get is 0.8 x equals 0.8 x plus their .4 Z. Well that will imply zero is equal to 0.4 Z. Which implies equal zero. But if Z is equal to zero, well that means that you don't even have a box anymore and therefore this one you do not have to use. And therefore from here we can say that if why in terms of X. This will be simply X. And then you repeat that for Z. So when you do the same thing, what we did for why using equation two and three we cancel out one of the excess And get Z is equal to two Y. But since Y is equal to X, therefore Z is equal to two X. And for the same reason This factor the lamb, the X -0.4 which applies the lamb. The x equals 0.4. Well that won't work because that implies one of the variables will be zero and therefore you don't even have a locks. And so from here. So after this use these facts, X equals X, Y equals X, and Z equals two X. To plug in into the cost formula. Sorry about that. So plugging in all these variables as X. Into the formula for the cost, what you get is this. And when you solve for X, X equals skirt five white was five and Z is equal to squirt five. Use these to plug it in back to the v. And we get is v. maxes 10 skirt of five.

But Russian tells me I'm gonna construct a box, and it's an open rectangular box. So here's my rectangular blocks. Um, bear with my drawing here is a little sideways, but it says it's gonna have a square base. So I'm gonna call this X and X, and I don't know what this is all gonna call it the height, and it tells me that the volume of my blocks is going to be 48 cubic feet. Okay, So that also allows me to say the volume in terms of its variables is gonna be X squared age. So that means that H is 48 divided by X squared. We're gonna use that later. Because what we're told is we want to minimize the cost of the box. Okay, The cost of the box is going to be since it's an open top. The bottom, which is X squared. This is the bottom cost six feet, $6 per square foot, and then four sides. Okay, so four x h is and they each cost $4 a square foot. So my cost formula is six x squared plus 16 x h. And then what? I could do is. I can say my cost formula then is actually six x squared plus 16 x multiplied by 48 over X squared. Okay, so that becomes sequel Six X squared. Plus. And I'm just gonna write this, um, as 16 times 48 over X for now, because I may be able reduce afterwards. Okay, but those X's cancel. That's what I'm doing to read one of those exits. So I'm gonna take the derivative. So see, Prime is gonna be 12 X and then minus 16 times 48 over X squared. Okay? And the question wants me to minimize that, so I'm gonna set it equal to zero. So I'm gonna say 12 X then minus 16 times 48 over X squared is equal to zero. So what I'm gonna do is say 12 X then is equal to 16 times 48 divided by X squared. Okay, so that means that X cubed is 16. Divided by 48 divided by 12 are 16 times 48 by 12. The X huge becomes and this is why I left this like that. That actually just is 45 12 just four. Look at four times 16 and four times 16 is four cubed. It's four times, four times four or 64. So that means X is gonna equal four. Okay, now, if X equals four, then I go back up here and I could find the height. So I say, then the height is 48 divided by four squared, which is 48 divided by 16 which is three in the box. The dimensions are four by four by three. Hey, that's my answer for the dimension of the box. And this is what is the minimum cost. And the cost again is the six X squared. We go six times minus, um, 16 times 48 divided by four squared. OK, though, that's gonna give me just six times 16 minus 16 times. And let me just do it this way. Um, this is gonna cancel 16 times four square are divided by four squared story is just cancel. So I get 48 so six times 16 minus 48. That's six times 16 minus three times 16. Okay, so that's going to give me three times 16. We're sorry, Ian, playing that into the cost formula, my bed, which is a plus. We're gonna add 48. So let's change this. That's nine times 16 and nine times 16 is 144. No, I need to fix that as well. Sorry I didn't fix this. It's not four squared down here. It's always a good idea to write the correct for. So let me just fix this it 16 times for anybody by force that's going to be 48 times four, we just three times 16 times for So that's 12 times 16 plus six times 16. So that actually gonna be 18 times 16? Which means this is gonna double. So my apologies for the lack of carefulness, but the cost comes out to 288. And if you didn't follow what I did there in terms of following my mistakes as well, if you go back up here, it's probably easier to just say I'm put X Um, in his four up here is a six times foreswear, about 16 times 48 divided by four. Okay, and then follow that down. If you're not sure, but ends up being $288

So in this problem, we are constructing a box that has a square base. Hey, and we want to minimize our cost. The volume that we want the box to be and it does not have a little on it. Just the bottom and four sides. The volume we want is 48 cubic feet. The bottom cost $6 per square foot to construct and the sides or $4 per square foot. So we want to minimize the cost of this. So our cost, um to calculate that we're basically going to use the area and multiply that times the cost. So for the bottom, we have $6 per square foot. And to find our square footage, we're gonna have W or with squared on the bottom. Now the sides are $4 per square foot. There are four sides, and the area of each side is with times height. Okay, now here we do have three variables instead of two. We have a C w in an age, so I want Teoh eliminate one of those. So what I'm gonna do over here, I know the volume is gonna be w square times age and I'm told that we want our volume to be 48. So I saw this for H and I get ages equaled 48 over W squared so I can go back to my cost formula, and I can plug in 48 over w squared for h simplify a little more. I'm gonna have six w squared and then plus 768 times w to the negative First power. And I put like them. I put it w to the negative first power because now we're gonna take the derivative, and I just find it easier to do that. So my derivative is gonna be 12 w minus 768 times w to the negative second power, right? And we want to set that equal to zero to minimize it so that I'm gonna have 12 w is equal to 768 over w squared 12 w to the third is equal to 7 68 W 2/3 is equal to 64 therefore w is equal to four. All right, so I know my width. Our height is 48 divided by W squared. So my height is going to be three. So my dimensions off my box are four by four by three. Okay, Now we want to know what is the cost of the box. So the bottom we know is $6 per square foot, so I'm gonna have six times w squared. So six times four squared, that's going to give me $96 for my sides. $4 per square foot. Times four. There's four of them. So I'm gonna have 16 times w times age to 16 times four times three Gonna give me 192. So my total cost for this box is going to be $288.

We're told that a cargo container in the shape of a rectangular solid Must have a volume of 480 cubic feet. Were asked to use the garage multipliers to find the dimensions of the container of this size. It has a minimum cost. The bottom will cost $5 per square foot to construct. And the sides and the top will cost $3 per square foot construct. So we're going to use the garage multipliers 1st. Let's figure out the cost. They want to minimize the total cost. C. This is going to be in terms of the dimensions of the box. Well, this is the surface area. This is eight times X times Y. Yeah, plus six times x times e plus six times Y times Z. In our constraint is that the product X, Y Z are volume Is 480. And now we'll use the garage multiplier method. So I'll take the partial derivatives of our cost. See, it's the partial of C with respect to X Is eight Y plus six Z. And we know that this is going to be equal to lambda times partial derivatives. Uh huh. Our function X. Y, Z -480. Which is simply Y times Z. The partial derivative of C. With respect to Why? Well this is eight times x plus six times Z equals land at times X times Z. And finally the partial derivative of C with respect to Z is six X plus six Y. And this is equal to land at times X. Y. Now let's eliminate Parameter Lambda from these three equations. So we have from our first equation since Y and Z. Or non zero. Lambda is equal to eight y plus six z over Y. Z. And the second equation, lambda equals eight X Plus six Z over XC. And finally Landed equals six x plus six wide. Uber X. Y. Mhm. Mhm. So equating the first two, we get eight Y plus six Z. Over Y. Z equals eight X plus six Z over X. Z. So okay. Eight X. Y plus six X. Z equals eight X. Y plus six Y. Z. looks like both sides by X. Y. Z. We're instead of doing that cross multiply. Mhm. This simplifies to six x equals six y. or simply X equals Y. Now with X equals Y. Plug this into our 3rd equation. We get lambda equals 12 X. Or instead of exits do why 12 Y over? Why squared? Which is simply well why? So we have this is also equal to eight y plus six z over Y. Z. Mhm. Mhm. And so 12 Y squared Z equals eight Y plus six Z. So we have 12 y squared minus six Times z equals eight. Y. Actually, yeah, going back to our initial equations, we have 12 Y equals lambda, Y square. Yeah, So this is actually supposed to be 12 over Y. And so we have 12 times Z is equal to eight Y plus six Z. And therefore okay we have six Z equals eight Y. Or three Z equals four Y. It's now fully eliminated lambda. Now you can simplify the two relations X equals Y and four Y equals three Z. And we'll plug these into our volume constraint so that we get yes. Four thirds why cute Equals are vol 480. Mhm. In solving for Y, We have why is equal to the cube root of 360? Mhm. And therefore it follows that our dimensions X, Y and Z Are the cube root of 360 Q. Bird of 360 and four thirds Times The Cube Root of 360. And these are all in feet


Similar Solved Questions

5 answers
Vf € RJa; b] . Ilflpfl
Vf € RJa; b] . Ilflp fl...
5 answers
Homework: 6.5 Homework Score: 0 of 1 pt 12of 16 (15 con6.5.71 Use an identity to solve the following equation on the interval [0,21)cos 2x = cosXSelect the correct choice below and, if necessary; fill in the answer box l0 complete your ch(Use integers or fractions for any numbers the expression: Type a exact answer There no solution;Clck select and enter your answerts) and then cllck Check Answerparta showingClaar All
Homework: 6.5 Homework Score: 0 of 1 pt 12of 16 (15 con 6.5.71 Use an identity to solve the following equation on the interval [0,21) cos 2x = cosX Select the correct choice below and, if necessary; fill in the answer box l0 complete your ch (Use integers or fractions for any numbers the expression:...
5 answers
Find matrix used to find the projection of Vector On to line making 1508 with the positive x-axis_ Use this to find the projection of the vector (1,3) onto that line
Find matrix used to find the projection of Vector On to line making 1508 with the positive x-axis_ Use this to find the projection of the vector (1,3) onto that line...
5 answers
8) (3 points) Find the critical numbers of f (if any}:f(z) =~V) 4points} Find the open Intervals on which the function increayinxdecreasineFirst Derivative Test identify relative extrema (2 points) Apply the
8) (3 points) Find the critical numbers of f (if any}: f(z) = ~V) 4 points} Find the open Intervals on which the function increayinx decreasine First Derivative Test identify relative extrema (2 points) Apply the...
5 answers
Complete ' the following truth table. (5 polnts)and ~qporor ~q
Complete ' the following truth table. (5 polnts) and ~q por or ~q...
5 answers
Question 26What is the pHof a0.010 M solution of hydrazine chloride; NzHsCI? (Kb = 8.9 x 10 7)2.759.974.0311.254.97
Question 26 What is the pHof a0.010 M solution of hydrazine chloride; NzHsCI? (Kb = 8.9 x 10 7) 2.75 9.97 4.03 11.25 4.97...
5 answers
An electron and proton each moving 840 km/s in perpendicular paths as shown in the figure igure At the instant they are at the positions shown in the figure.Figureof 1Electron5.00 nmProton4.00 nm
An electron and proton each moving 840 km/s in perpendicular paths as shown in the figure igure At the instant they are at the positions shown in the figure. Figure of 1 Electron 5.00 nm Proton 4.00 nm...
5 answers
Figure 1-1. Typical chemical plant layout By-productsReactorSeporotorRov MaterialsProductsRecycleSoparation and PurificationSeparators
Figure 1-1. Typical chemical plant layout By-products Reactor Seporotor Rov Materials Products Recycle Soparation and Purification Separators...
5 answers
Question 12 Not yet answeredMarked out of 2.00Flag questionA double-convex lens has surfaces whose radii of curvature are both 40 cm. The index of refraction of the glass of the lens is 1.6. What is the focal length of the lens?A. 33.3 cmB. 12 cmC. NoneD. 3.33 cmE.20 cm
Question 12 Not yet answered Marked out of 2.00 Flag question A double-convex lens has surfaces whose radii of curvature are both 40 cm. The index of refraction of the glass of the lens is 1.6. What is the focal length of the lens? A. 33.3 cm B. 12 cm C. None D. 3.33 cm E.20 cm...
5 answers
Xhas normal distribution with the given mean and standard deviation_ Find the indicated probability: (Round your answer to four decimal places_ p = 95_ 0 = 15, find P(110 <X < 134)
Xhas normal distribution with the given mean and standard deviation_ Find the indicated probability: (Round your answer to four decimal places_ p = 95_ 0 = 15, find P(110 <X < 134)...
5 answers
The enzyme ornithine decarboxylase generates a product that has been proposed to play a role in stabilizing compacted DNA. Draw the structure of the ornithine decarboxylase reaction product and explain how it interacts with DNA.
The enzyme ornithine decarboxylase generates a product that has been proposed to play a role in stabilizing compacted DNA. Draw the structure of the ornithine decarboxylase reaction product and explain how it interacts with DNA....
1 answers
Determine the reactions at $A$ and $B$ when $\beta=80^{\circ}$.
Determine the reactions at $A$ and $B$ when $\beta=80^{\circ}$....
3 answers
'ilrn/takeAssignment takeCovalentActivity do locatorzassignment-take TReletenceal TUTOR Relative Reaction Rates and StoichiometryFor the decomposition of hydroget peroxidc dilte sodium hydroxide at 20 'C2H,O-(aq)H,o() O_(2)thc avriage rale Of= disappearance of H-O: DlCi thc time period fTOmU I0 [453 min found bc 214*10" MminWhat I5 thc &vcrage rate of appearance of 0; Over thc same Timc period?Lin5hon AnnmacnShow Tutor SiCdsSubmtit
'ilrn/takeAssignment takeCovalentActivity do locatorzassignment-take TReletenceal TUTOR Relative Reaction Rates and Stoichiometry For the decomposition of hydroget peroxidc dilte sodium hydroxide at 20 'C 2H,O-(aq) H,o() O_(2) thc avriage rale Of= disappearance of H-O: DlCi thc time period...
5 answers
16. (a) For the Linear Price Model Pn+l 18 0.8 Pn, find the largest value of Po for which Pn > 0 for all n 2 0. (b) For the Linear Price Model Pn+l 18 1SPn, is there a value of Po for which Pn Z 0 for all n > 02 If so, find one_
16. (a) For the Linear Price Model Pn+l 18 0.8 Pn, find the largest value of Po for which Pn > 0 for all n 2 0. (b) For the Linear Price Model Pn+l 18 1SPn, is there a value of Po for which Pn Z 0 for all n > 02 If so, find one_...
5 answers
300.0 mL of a 0.340 M solution of NaI is diluted to 700.0 mL.What is the new concentration of the solution?
300.0 mL of a 0.340 M solution of NaI is diluted to 700.0 mL. What is the new concentration of the solution?...
5 answers
Copper wire has diameter of 1.697 mm; What magnitude current flows when the drift speed is 2.09 mm/s? Take the density of copper to be 8.92 X 10 kg/m
copper wire has diameter of 1.697 mm; What magnitude current flows when the drift speed is 2.09 mm/s? Take the density of copper to be 8.92 X 10 kg/m...
5 answers
For the reaction below:1. ether Hzot2 PhMgBrOCHzCH3Draw the major organic products.Draw one structure per sketcher: Add additional sketchers using the drop-down menu in the bottom right cornerChemDoodle
For the reaction below: 1. ether Hzot 2 PhMgBr OCHzCH3 Draw the major organic products. Draw one structure per sketcher: Add additional sketchers using the drop-down menu in the bottom right corner ChemDoodle...

-- 0.022173--