in-the-following-exercises-solve-the-given-maximum-and-minimum-problems-a-beam-of-rectangular-cross-

Question

In the following exercises, solve the given maximum and minimum problems.
A beam of rectangular cross section is to be cut from a $\log 2.00 \mathrm{ft}$ in diameter. The stiffness of the beam varies directly as the width and the cube of the depth. What dimensions will give the beam maximum stiffness? See Fig. 24.67.

Sherrie Fenner · Accepted Answer

all right, Question 30. I think we need now we have s balls, K h squared of you, and we have a diameter equals 20 inches. And we, of course, know our factory in here. Um um, Palm Beach, where plus the rooms for years equals 20. Weird, because we&#x27;re dealing with a log attempt to draw this, You my love. So I&#x27;m drawing my triangle in which my hi my diameter is actually my, um, iPod news. You make that out there. So I saw my, uh, equation for H. And I get each this very, uh, negative Davies player plus 400. I&#x27;m gonna substitute that in for each in this equation. No, I had as in cool sometimes. Didn&#x27;t. Kay Negative. Devon Square is actually a member radical waas 400 and squared. So the radical sign and we&#x27;re gonna cancel each other out. I&#x27;m w when that I am s e equals k time. Negative use were plus 400 time W I&#x27;m gonna distribute that w may have you Q plus 400 that you Now I&#x27;m gonna take the driven, and the constant would actually stay as a constant. So I&#x27;m gonna write my derivative without the constant. But I know that the constant would still be a multiplier. So, yes, I&#x27;m sorry that zero Vero equals negative Read of use squared plus 400 and then in itself for better you negative 400 equals They negative three w square. So my signs cancel divide by three or 100 over three. It&#x27;s weird that I&#x27;m only walking talking about a distance. I don&#x27;t have to worry about the negative equals W spirit of 400 is 2020 over this square root of three. Gonna rationalize that denominator. And I&#x27;m gonna get 20 square root of three over three. And then I&#x27;m going too. So that was for W. Plug it back into my equation or each so I had huge. You won&#x27;t negative and just gonna do you root them or 100 over three square plus 400. So this radical and the square cancel out and I have a TSH equals 400 degrees, 400 once for grand. That would be H E rooms. I won&#x27;t. Every 12 hundreds have 800 100. The worst will be one times two over this food for three and we rationalize that we would get ht equal money times were six

Ralf Melis · Answer

okay. Little work problem. We have a rectangular being, and then that&#x27;s to be cut from a cylindrical long. And we are told that the diameter is 20 centimeters. So in that picture here Damn, it would be from here to here. That would be 20 centimeters. And question one says expressed the cross sectional area as a function of angle data. Okay, so, uh, we&#x27;re gonna assume that we have right angles here. And when we do that, we can write little equations for signs and co signs. So let&#x27;s start with the sign of data. For example, if you start with the sign, then that would be opposite over high Parton&#x27;s in right triangles. So this is the depth divided by 20. If I write the coastline here, then I would have the with which would be over here as well. So that&#x27;s the adjacent side. Over I apartments, niches 20 again. And then the reason we&#x27;re doing this is because the area of this rectangle would be equal to be depth times away. So I&#x27;m just gonna rewrite these so that been solving for depth and for with and then I&#x27;ll just plug it in. So we&#x27;re gonna cross multiply here, sort of about 20. So think of this is being over one. You cross multiply, you will get that to depth is 20 times. Sign data. I do that over here. Think of this is being over one so that when you cross multiply, you would get to the with is equal to 20 times the co sign of state, and then you can plug it into the area formula, which is depth 20 Sign data times. With that, she&#x27;s 24 sign beta. And so we get an area expression that it&#x27;s 20 times to wanting that 400 times the sign data co sign data. Now you can leave that, Um, but it would probably be easier if we rewrote this one more time because we have identities that we can work with. And there&#x27;s an identity that would work well here. And that would be that we can rewrite this as 200 times two times the sighing data co sign data because to sign beta co sign beta is actually equal to resign to fate. Um, and by doing this, I just dealing with one. You&#x27;re gonna metric function instead of having to that will make my life a little easier. Okay, so there you go. That&#x27;s a so be says, I need to grab this. So let&#x27;s make this a little smaller so that we have a lot of my room. So we&#x27;re gonna graft this. It didn&#x27;t tell me that I could calculator or not, So I can&#x27;t. So, by graphing is, uh, equation. I&#x27;m gonna identify a couple things here. So first of all, since 200 years for the amplitude. So that would be how higher low the science sort of goes. And then because there is a two data in here, I know that normally the period for the sign of failure is two pi. So the new period will be to pie divided by and then reuse that, too, and then to part of item by to use high. We normally split to sign up into four equal parts. So I multiplied by 1/4. Then that means in my table I&#x27;m going to count by pi over four. So what I&#x27;ve done at this point that I could make her a table or data I&#x27;m going to start. What state of being zero on account by power before. So one pile before to power for that should be pi over two three files for it&#x27;s It&#x27;s not reduced in the four by over four, which is pie. Good, because I said my period would be high, so I got back back. And then I have this line that I&#x27;m not compelling because I know that normally designed starts at zero upto one compactors there are bound in negative one on girls back zero. And then there&#x27;s amplitude here. It&#x27;s just a multiplier sort of stretches still function. So a multiplying bicycling bit zero times doing it 01 times 200 was 200 0 negative. 200 and they&#x27;re over. And that&#x27;s all the information that I need a spot. It&#x27;s graphic this You have to excuse my little squiggly graph here. So, um, you know, it&#x27;s periodic. Don&#x27;t do with what happens on the left side here a little bit. So we&#x27;re gonna go what? 123 four steps out. Uh, best pine. I need to be able to go up to 200. So clearly that&#x27;s a different scaling on the X and the Y axis. So we&#x27;re gonna start at zero. I&#x27;m gonna go up to 200. That found zero. We&#x27;re down to 200 single, best up to zero. So the bad in here would be that one tick marks to the left. It should be down there. So let&#x27;s raise taxes when good, it&#x27;s in the way. No, put a dot right there. So I will put the negatives to get over here. And then that&#x27;s enough to give me a graph that looks like this. So this is what it looked like if it was a function. Um, now it is a word problem. Let&#x27;s see if I really need all of this. So, um, considering that this is a work function, actually that views, uh, an area and angles. It probably don&#x27;t need this part over here. So because it doesn&#x27;t make sense that negative areas, right? And so I would probably agents. You can leave the graph like this so that we just have positive angles. Was fun. That&#x27;s stated when you want to use to make that positive or remain even just sort of like more this because that service, Ingrid, that we would have a negative, right? So it&#x27;s it&#x27;s one of those two, but there you go. All right. Last quite shooting. Says, find the dimensions of the being with the largest cross sectional. Okay, so from the graph here, we can see that the largest cross sectional area, the maximum occurs over here. Right. Um now I know that that happens. That power before based on my grant, I also know normally I didn&#x27;t have the graph. I couldn&#x27;t reason my way through that as well, because the maximum value you could get over out of any sign. So the sign of anything, The sign of Alfa the maximum value is always going to be one. It doesn&#x27;t matter what they say. So we haven&#x27;t to fate of there. It doesn&#x27;t matter in this case. And that a curse at 90 degrees. So the sign of 90 degrees is one. Oh, if 90 degrees would be one. So then Alfa would be 90 with ships pi over too many l for its tooth data. It would be too faded. Spy over to baby. It&#x27;s piled four Beautiful back two favors firewood, Endemically divide by two. You could have solved for pile before as well. So you can see there to readies to get pile before. So now we just need to find the maximum the mentions. So I had the depth. I still have this from questions May I said the depth is 20 times the sign data 20 times, Sign of beta and say that they is filed before and 20 times. Sign of data. So before on circle Power four would be over here, so that would be 112 It&#x27;s a 45 degree triangle, so the sign would be won over two. Okay? Or maybe you have that memorized. And then we could rationalize that the nominator by multiplying by two over route to so 20 route to over 13 which is 10 route to. And if we were allowed to use a calculator, um, we could put this in, we will get our and set and then the way, um would be saying, because we would have 20 times the co sign filed for and the constant fire of where he&#x27;s also won over to. So that would be exactly the same. So both of these would be identical to each other, which is interesting, because that implies that this is a square. So the maximum area looks like it has something to do with being sweat. Great. So you guys later

Regina Hays · Answer

going to start with a, um, circular log that has a diameter of 12 inches. And we want to cut out a rectangular being from this log, and it&#x27;s going to have some kind of with in some kind of depth. Um, and we want to maximize the stiffness of the being, and we&#x27;re told that the stiffness is actually proportional to the were times a cubed of the depth came. So that&#x27;s how we calculate stiffness. And so, since we want to maximize, I know we&#x27;re gonna have to take the derivative and we don&#x27;t want to do it. K is just a proportionality constant, so he doesn&#x27;t come into play taking the derivative, but we still don&#x27;t want the stiffness depended upon two variables, and so we know that 1 44 or 12 squared is equal to W Squared plus D squared and let&#x27;s go ahead and solve four w. And so 1 44 um, or sell for D minus. W squared and take the square root of that is equal to D. And of course, W is going to be constrained between zero and 12 inches. And so my stiffness is equal to K times W times 1 44 minus W square to the three halves power. And so, unfortunately, we&#x27;re gonna have to use the product role to take the derivative. Um and so the derivative of my stiffness with respect to W. Is going to be K times at 1 44 Money&#x27;s W Square to the three halves plus k W times three halves times 1 44 minus W square to the positive 1/2 times negative to W And let&#x27;s go ahead and simplify this a little bit. So we have K times this 1 44 minus w square to the three halves. Um, the twos cancel out. So we have a minus. Um, que three k w squared times 1 44 minus w square to the 1/2. Okay, we noticed we can factor out of K. So we have set that equal to zero because we&#x27;re looking for the critical values and I&#x27;m a factor at K, and I&#x27;m going to factor out a 1 44 minus w squared to the 1/2. And when I do that, I&#x27;m love with a 1 44 minus w squared minus a three w squared. So this is going to be K times 1 44 minus W square to the 1/2 times a 1 44 minus four w squared. And so, um, this part gives me a W of 12 inches, which is one of my in points, and this part gives me a W equal to six inches. So now I want to know where does my maximum strength or magnet, not strength that stiffness occur. And so, um, evaluate my stiffness at zero. And when I do, I get zero my stiffness at 12. And once again, I get zero. And in my stiffness at six. And when I do, I get six k times this 1 44 minus 36 to the three have now. I don&#x27;t necessarily care what it ISS other than, um, yeah, I do. I do care, cause I&#x27;m not taking the second derivative. Um, and so this is about someone. Go ahead and stick it all in my calculator. 67 34.21 times k. So this is my maximum stiffness. Look at stiffness, um, at w equal to six inches. And then if I can solve for my depth, I get six times a screw to three inches. Okay, Now, what I want to do is I want to verify this graphically. So I&#x27;m gonna let k equal toe one. And so I want a graph. My stiffness, Um, with respect to W in case one. So I get, um, w times 1 44 minus w square to the three halves. Um, and so I may change to a graphing utility, and I&#x27;m gonna go ahead and graph. Um, and of course, I have to change everything in terms of exes. Who this was Rex kinds 1 44 minus x squared. And then, of course, that is raised Club may take it, of course, is raised to the three house power. So that is what the graph looks like. Um, and thankfully, I had everything set. So, um, 0 to 12 just by, um, sheer Let, um I know my with goes from 0 to 12. That&#x27;s a maximum. And of course, we knew that our maximum, um, maximum stiffness was 67 something 67 34 by our calculations. Anyway, So we can set our window and you notice our maximum value does occur when w six and a maximum value of stiffness of 6700. Okay, so that verifies it. So let&#x27;s go ahead and go back and let&#x27;s go ahead and do a quick, um, sketch from here to here. So this is 12 and this is appear in about 6800 Let&#x27;s say and so if this is six. So we did have it come up like this. And so that value was at six comma 67 34. So we verified what we solved for through the derivative. Now what we want to do is in the same window. We want to graph the stiffness, depending on the depth instead. And so if I go back and solve that, I get a depth cube times the square root of 1 44 minus D squared. So let&#x27;s go ahead and graph that in the same window. So let&#x27;s go ahead and growth that in the same window to now I have G of X equal to X, and that was cubed times the square root of 1 44 minus x weird. So that is what that graph looks like. And so if I come over here, I do know that the depth was six times a screw to three, which, if you put that you calculator, that is about 10.3 10.4. So that validates that as well. So I may come back here and over here I know that this graph actually looks something like something like that. And so this now is about 10.4 comma 67 34 thereabouts. And so that once again validates what we determined through calculus. OK, so now what we want to do is figure out OK, we did this for women. Are proportionality constant? Was one was gonna happen if we change that proportionality Constant? My hypothesis is that is not going to change the values of W and D. It&#x27;s on Lee going to change the maximum value that those two that that where WMD are. So it&#x27;s just going to change. The 67 34 is not going to change the six in the 10.4. So let&#x27;s go back here. Um, and let&#x27;s go ahead and change. Um, let&#x27;s go ahead. And we can even have a proportionality constant of 1/2. And so you notice it did shrink it half the maximum value. But it changed didn&#x27;t change the w and so or the place where the maximum value occurs. And I do the same thing over here for my that equation. So the same thing. This is still 10.39 It doesn&#x27;t change the value of job une de it only changes the value of the maximum value. And so I can even go hair and double it. And so that is just to me with just trouble, the maximum value. So instead of it being 67 it&#x27;s going to now be at 13 something. So let&#x27;s double it, Um, and let&#x27;s go ahead and soon out. And sure enough, um, didn&#x27;t change the 10.39 It&#x27;s just doubled the 6700 or 6400 whatever it waas and so the same thing here, it&#x27;s still gonna be at six. It&#x27;s just going to double it. So if, um by changing by changing okay, it only changes the maximum value, not w anti where Max occurs

Angela Guo · Answer

we were given formulas for a W and H of which are with and height of the strongest two rectangular being that can be cut from a spoon trickle cylindrical log our any is the radius of that log. So for this question were asked to find the with the height and the cross sectional area of the strongest thing where the law that were working with has a diameter of four feet. So if the diameter is for then that means the radius, eh is equal to half the diameter which is to so that means W is going to be two times two over three times three power of one hand. So punch that into your calculator and you will get this is roughly grounded. The nearest 100 is going to be 2.31 and then the height it&#x27;s going to be two times 8/3 to the power of 1/2. Listen, they will punch that into our calculator, and we end up with This is roughly three point to seven feet. So this means a cross sectional area is going to be the width times height, So multiply them together, then you should get roughly seven point by four. 30

Tony Ni · Answer

So this question we&#x27;re being told that the maximum weight over but the returns would be support. So wait very strongly as its will and spirit of its height. English of its wings. Oh, that&#x27;s okay. Times with we&#x27;re a bit tight square anywhere. Savitt playing so one over. So that&#x27;s it. Whether that is okay, the new Age spur over l. Okay, so no, nothing, no. If Beam is one white, but one certain foot high and 10 foot long it supports told Hunt. So he goes to K A profitable one. Okay, I went over to I&#x27;m when the 1st 1 over certain No won over certain street three. Where over that hope along. So that&#x27;s home. So now we can complete. Okay, so he gets on my own terms here. That&#x27;s kept that crystal Kaye hang for number two times. One over C squared is one that we&#x27;re over, child. Oh, first thing we do is multiplied the 12 over. So that&#x27;s 12 times told. That&#x27;s 144 person Carrie Kind one over 18 that my both have anyone over 18. So that&#x27;s kid, because 144 times 18. So that becomes 25 to so then very close to Sonny by me too. Worst I&#x27;d over. The next thing we&#x27;re gonna do is next part of the question I asked. What is the way? If the being this too, sir. To Overstreet, foot wide one over 16 foot long. Been a pleasure. So they chose to 592 times over three times over, too, over exchange. And if I&#x27;m president going to crack later. Passion. And that was party chief 0.5 points, and that&#x27;s my answer.

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In the following exercises, solve the given maxim…

Problem 48 Easy Difficulty

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Samuel H.

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Precalculus Review - Intro

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Basic Technical Mathematics with Calculus

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

Catherine R.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus