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Atrough for holding ~ water for dogs at a local animal shelter is to be formed by taking = piece of sheet metal 60 Cm wide and 1m long and folding the 20 cm on eith...

Question

Atrough for holding ~ water for dogs at a local animal shelter is to be formed by taking = piece of sheet metal 60 Cm wide and 1m long and folding the 20 cm on either end up as shown below. Determine the angle that will maximize the amount of water that the trough can hold. Make sure YOu show your work for full marks.20 cm20 cm20 cIn

Atrough for holding ~ water for dogs at a local animal shelter is to be formed by taking = piece of sheet metal 60 Cm wide and 1m long and folding the 20 cm on either end up as shown below. Determine the angle that will maximize the amount of water that the trough can hold. Make sure YOu show your work for full marks. 20 cm 20 cm 20 cIn



Answers

A rain gutter is made from sheets of aluminum that are 20 inches wide by turning up the edges to form right angles. Determine the depth of the gutter that will maximize its cross-sectional area and allow the greatest amount of water to flow. What is the maximum cross-sectional area? (IMAGE CAN'T COPY)

Already. So for a number 43 We're talking about a trial and talking about volume. So the volume is given to be uh length times width, times height and were given in the dimensions that it is um 12 -2 X. Times X. Times 20. And so basically we got 20 X. That's going to get distributed. So that's going to get you 100 and 40 X minus 40 X. Squared. So rewriting that, that's gonna be negative 40 X. Squared plus 140 X. As our volume. So that's part a part B asks us then to find the maximum volume. So that's gonna be um the maximum amount of water that you'll need. So that's gonna be the opposite of B over two A. So that's going to opposite of 1 40 divided by uh two times negative 40. So that's going to be yeah Up to 40, not 1 40. I apologize to 40. There we go. Alright so negative to 40 over uh negative 80 is going to cancel out those zeros and leave us with three inches. So then in order to get the actual maximum value plug it in. So we've got negative 43 squared minus 2 40 times three. Okay You go and do that math, you're gonna get negative 360 plus 7 20 And that's going to leave you with 360

So for Question 96 a tin can is filled with water with water to get the 39 centimeters. The whole 11 centimeters above the bottom of the can produce a stream of water that is directed at an angle of 36 degrees above the horizontal. Find a the range and be the maximum height of this street. You can see that, too, wants the end of the chapter. We really still getting questions that are a little bit more interesting. I want to show you a little bit more about general physics on some more broads, aspects that you might get other than just the questions you see in this chapter. So recovering a broader range of topics. So we have an angle. See to their the drawer anyway, Let's just do some physics on this question on. We can easily find an initial velocity off our water by using Torricelli's equation, which states that the velocity off the magic water is going to equal to routes to H just a balance of potential kinetic energy. Gravitational gentlemen, Kevin since is equal to times 9.81 times this height and, well, just consider the height from what the questions say's. Although the can is filled with water to a depth 39 centimeters, the hole is actually 11 centimeters from the bottom, so it's not completely at the Boston. So the actual height difference that we're talking about 0.3 was attractive 0.1, which gives us our true height that we're interested in the height from the top off the camp from the field, and this velocity we can figure out is to work 344 It is a second. Okay, so now we need to right? Essentially an equation that governments how the articles of make or water think it's water in expressing milk with last week are going to fall. So a position off any of this is equal to an emission position to mission velocity. Most ignite by time, attitude, half times, acceleration times T squared or this taking place in one direction. But she's just fast, so we know that zero is equal to 0.11 That is, that your eventual position has to be on the ground but start 11 sentence. It's up onto that your initial velocity, which of course, is to want three full. Full. Good. Remember to not apply this by the sight 36 degrees as it's in the vertical direction. Most black party. Thankfully, this is rough. Roughly the hardest part of this, as the acceleration that was is a comfortable 9.8. So it's half 9.81 teams squared. What do we have here? Well, we haven't quite drastic because cord graphics quite often require the use of the quadratic formula. I'm gonna have to use another page just to express the whole thing. I like it the time that it takes full the water's land, it's going to be equal. See if we go back to the previous place just for a second. This represents, uh, b. This represents sea. And this represents out, eh? In the question A square. Hey, a X squared must be X. But see Izzy quiches era. So that's a quadratic. And we remember that was sold quadratic to say the X is equal to minus plus or minus the square root off B squared minus for a C and the entire thing to a on which is going to use the numbers that camp separately, work into a calculator here and used them in the core traffic. So minus B baby hotel on 0.37 minus 1.378 we'll take here, plus and minus squared off one point 378 squared minus four times in brackets minus four. Put 905 That is half off night for a walk, most by that by syrup for one. Those brackets. Yes, I know this looks particularly complicated, but using Cortright should to terrify, so divide the whole thing by minus two turns four. What? No. Five. Right. Let's get critical with this equation for a second because we can make things a lot more straightforward for ourselves. Just by example, What do we know? We're going to get two solutions and the square roots, all of the governs under the square root is going to be a number that is quite large. And we know from the fact that we've got two minuses in there that is going to be larger, that the evaluated square root is going to be larger than the magnitude off the negative. Be part I that is this YSL greater than this. And so what that means is that we've essentially got two solutions, one negative on one positive. And the only woman concerned about is the positive solution. But the bottom of the fraction here is negative. And so the chops should also negative so that the negatives cancel out. We get the positive solution to the fraction so we don't have to waste time working on both solutions. We can just simply evaluate solely the negative solution. And if we do this, if we put all of this in to calculate correctly, which actually will have practice with get a solution zero quake through for six seconds. Okay, great. So go back to the question for a second. And now what we're looking at. Well, if we're trying to find in the question the range on the maximum height, we're also going to want to continue doing more kin O Matics equations to understand the motion. Everything here? AII that is. How far does this milk this water goat? Question? How far does this wall to go away from the camp? The next thing we're interested in is this distance here rights. We're going to do this now, and we're going to require another page just so we've got room to work with this question. Thankfully, there is no acceleration in our ex direction, the horizontal direction and so the distance that Ugo is equal to the velocity at which Ugo and the time it takes for you to land with you just worked out. And so we know that our velocity of the ex direction is 2.344 lost. It worked out, multiplied by the coast site 36. And we must let that by the time it takes. She's no important 346 giving us distance off no 0.66 And here's our range. This is how far that girl's. So what about the maximum height? Well, let's consider back in the wind direction. That's a tie like this is the solution to the rich. Okay, for the maximum height, this is another straightforward question. But I'm gonna use a whole page of inspecting over a couple of pages. So one small beg forgiveness that I'm using yet another clear page. What do we know? Well, we know that at the top of its climb, its velocity is going to be equal to zero. So as the water balls in this path here. It's vertical velocity in Syria, and this is the key information we're gonna use to find the maximum height. So we use it close to your question. Is this so your final velocity anywhere on this situation, it's gonna be your initial velocity, most blood by scientists. So we're only taking the wind direction. And from this, we're going to subtract the acceleration g times the time it takes to get the top, not the time it takes the lab. That's very important. And when Mr Losses equal zero, we're going to find out exactly how long it takes to It's okay. Once we can plug plug that value of how long it takes to get to the top, we can plug back into the height of question and we'll find out what their high cheese. But for now, that's so for the time it takes to get it up. So is equal to your initial velocity most, but by sign, that's six. I would divide by G through level these alleys in 2.344 signed 36 divide by 9.81 I'm getting that 0.140 seconds. I'm now into the closing statement. You know that the height and get point get equal to some initial height. You gotta remember that it starts. Know quite what one set, one more meat is about the ground. Plus some initial velocity by the sign 36 times time. It's obviously time taking chances. So, you know, this is tight top. I'm will abscond with the tractor. Oh, times d Times t squared T top spread. We have all these valleys, so we're gonna do warn. Great big substitution started. No one said that means is above the ground. You traveled initially 2.344 Sign 36 Most blood that by no pipe once Lizzie seconds subtract from that half. So it's 9.81 times no forms 140 seconds. And now it's time to put that into your calculator. Sort that has to be squared. They make that stick. So what value do you get? Get what I want is equal to north points to one meters. It is the height cheese. See, that is your range no 0.66 meters and your height No point to one beaches

Okay we're looking at maximizing I was given a practical problem so we have a shape of our menu and we know this is 20 inches wide from here to here. 20 and then what we do as we fold the corners so we get shaped like that the square missing the topic And now we know that Islam is still 20. However we have three sides to it now. So we know these sides are equal. We're going to call them both. Why? And then we have this length here programs call mhm And we know that we must have Uh huh. Was to loss of why Must be out 20. So that's our restriction. Now we want to maximize the surface area which is this area here I received this is given by X times Y. So ask for this area. Okay is equal to mm times why? Just the red area. So this is a function we want to maximize, we want the maximum possible surface area. So what we do as we're going to rearrange this equation in terms of act we have back 4 20 minus two why? And we're going to substitute that into here. So our equation for area just depends on one of our blunt. So what this becomes is a Equal for X is 29 2 y. And my wife. And then we can expand this 21 -2 Y Squared. And then what we're going to do is we're going to horizon that would take out a factor of two Y. Then we'll actually and minus why and then we can see that we can draw a scab over here. And so we have over here changing values of life. And then the corresponding every values for it. We see there's a root here at wife was zero And that life was 10 and we noticed that the coefficient and the y squared term is negative. So it's currently starting in this conference and anything in this one. And the graph was something like this. So we can see at some value of why over here a function of a reaches a maximum. This is what we want to find. You want to find when this graph reaches maximum. What we're going to do is we're going to rearrange this quadratic and we're going to completely square find where this turning point is. If we take out 5 -2, y squared minus 10. Why? And then carrying on completing the square, Yeah. Why minus five? We're minus 25. I'm not mrs equal minus two Y minus part. Where at mm 50. And so we know that this turning point because at the point 5 50 marina from here, you hear what the of the why and the egg coordinate will think. And so if we know that and maximum occurs that y equals five, then we go back to our equation for X. Here, we substitute them. The y equals 20 -2. Lots of why time X. It was time. And so we found out too long. Why? Yeah, is equal to five And X is equal to 10. And so that's a maximum maximum values needed to maximize the surface area. And so our search there is given here which is X times Y. They won't survive times time, just 15. And so we conclude, yeah, the maximum mm service area was given by ACT it was can 585 and has following you 15. And each unit is images square and these both inches.

We have given a cushion are the injuries made from a seat or aluminium that are 2020 inches wide. So from portion began simply by the girder. So we can go this equation as 20 minus two x We can see that this is the land or we can replace this one s They replace X with Grandin minus two x and away with X in a video that is X right, So we can simply right area is equal to going T minus two x in two X and when we're solving this one So we're getting area is equal to 20 X minus two x squared so we can simply write this one and what we need to do. You were just comparing this equation with the standard equation and then we will find who were takes here. So forward takes begin Simply say darted. Maybe we'll do minus 20. Mind minus doing todo so it will be well to fight Well, it is a little X here, so this is a festival. So from here we can simply write that the value off way when we're boarding this Well, you will also be ableto or we can simply say that the length well, then equal toe length physical So we can simply say 20 minus doing to fight or it will be called to stand in just and area will be maximum area will be 15 square inches.


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