5

Cylindrical tank I0 f. high with rdius sf . half Glkcd with & Qluid having 64.32 Wb/ft. Sct up an intcgral to find thc work done in pumping thc fluid out of the...

Question

Cylindrical tank I0 f. high with rdius sf . half Glkcd with & Qluid having 64.32 Wb/ft. Sct up an intcgral to find thc work done in pumping thc fluid out of the tank: (10 points)

cylindrical tank I0 f. high with rdius sf . half Glkcd with & Qluid having 64.32 Wb/ft. Sct up an intcgral to find thc work done in pumping thc fluid out of the tank: (10 points)



Answers

Two very large open tanks $A$ and $F$ ($\textbf{Fig. P12.83}$) contain the same liquid. A horizontal pipe ${BCD}$, having a constriction at $C$ and open to the air at $D$, leads out of the bottom of tank $A$, and a vertical pipe $E$ opens into the constriction at $C$ and dips into the liquid in tank $F$. Assume streamline flow and no viscosity. If the cross-sectional area at $C$ is one-half the area at $D$ and if $D$ is a distance ${h_1}$ below the level of the liquid in $A$, to what height ${h_2}$ will liquid rise in pipe $E$? Express your answer in terms of ${h_1}$.

Were asked to find amount of work required to pump out half the half of the water if the tank started off. So the first thing we have to figure out is how much exactly is half of the tank? So we define the full capacity of the tank, and we're going to use the formula for wall in with a calm, which is won over three high H R squared. So in this case, were given our ages 10 and R R is five Bill, which means this is going to be won over three times. Pi times 10 I was 25 so a full tank is going to have a bawling of 2 50 over three times, so this means half of the volume is going to be half of that. So it'll be 1 25 over three times. And we wish to find, um at what height does that happen? Because that is going to help us with our own lives of integration leader. So we know when to stop. Let's do that. We're going to do a similar triangle, so let's say we have a cross section of the cone and we take half of it. So this is our radius, which is five. This is our height, which is 10. Well, let's say our water is here. That's going to be Are this is going to be I'd say h thank you. We know that, um are over. Age is going to be fly over time or are is going to be 1/2 time speech. So knowing that we can now solved what are sorry Sold for H So that's going to be won over three. Thanks, Pai. So the height is going to be capital H and R squared. That'll be one over to h. Great. Cancel out the three pie. Like it. 1 25 is equal to h cute over for. So that means H is going to be equal to bye 100 to the power of one. Okay, so that means we're going to stop when the height of the water. 500 power of one over three. So now we can start calculating, um, the amount of work it takes the pump out each sheet of water, let's say, would have a chute of water here. So the volume of that sheet of water is going to be equal to Pi R Squared, which is the area of the circle and times Delta y, which is the height of on a sheet of water that's going to be high, are square delta y and our We can calculate that in terms off in terms of why which is the height, some of the water. So, in fact, we have already calculated that here, you know that with similar triangle, the radius is going to be won over two times each. But now, instead of using a TSH, we're going to use why. So we're going to assume that the height of the water is equal toe. Why, then our is going to be half a wife, So that is going to be high planes. Why over to all squared dying? Don't tell why. Next we're going to find the mass, which is that city times volume. So density is 1000 and then we'll factor out the the 1/2 squared. But 1000 times 1/2 scored, I'll be 2 50 This is 2 50 times pi times life squared. I'm still tell why next we'll find the force, which is mass times gravity so that is going to be equal with the true 50 times 9.9 point eight, which is going to be 2450. Hi. Why Square Delta y now? The last part is finding the amount of work it takes to pump out that sheet of water. So work is going to be the force times distance that we have to move it. So we're called out. I said, Why is equal to the height of the water than diamonds? A distance? That it has to travel is going to be Tang minus y, plus the length of the spell. So which means the total night that has to travel is going to be 12 minus. Why now? We can put it all together and calculate the total work which converges to an integral, and it's going to be, um, four times told finest wife. So that is going to be 2450 Who's high times y squared times 12 minus y. Why? And now we have to find her limits and integration, which is where this number comes in. So at the top, our wise going to be equal to, so that's going to be our upper. And then the bottom part, um, when we stop this when h is equal to this number here. So that is our lower limit. 500 to power. So which means this is going to be equal to. So we will take out or Constant now, 503. This town this is going to be 12 y squared minus. Why cute. Why? And that is going to be equal to more. Why? Cubed minus one over for wide of the power of four. Evaluated from 500 Power one over three. What time? So we're gonna punch that into our calculator. Okay, so after punching all this into the calculator, you're going to get three 0.7. Wait seven No millions have under 87,000 100 and 42 jewels.

The question is, a water tank is in the form of right circular cylinder With the height of 20 ft And the radius of six ft. If the tank is half full of water, find the work required to pump out, uh pump out all of it over the top. So now moving towards the solution, work done is calculated by integration from me to be F of X. Dx. Now let X be the distance measure from the bottom of the tank to pump the layer of the water of thickness deer legs at X feet From the bottom of the work needed to be done as 62.4 By 6 m20- X. Dell X. Now, by the definition we can find the work done is work done equals to integration from 0 to 10 36 point 6245 20-6 d. x. So now moving uh Solving this integral, you will get your answer as 1058 591 feed. Help with. Thank you.

All right, so let's start out by drawing a picture of this cylinder. So it's a cylinder with height, eight meters and radius of three meters. Eight. We have our radius of three meters and were asked to find the work that is required to empty the water in the top half of the tank. So let's write down our handy formula. Work equals the integral off pressure times gravity times A cross sectional area multiplied by in this case, the distance off the water that has to be lifted. Um, okay, so let's set our boundaries well with the top half the tank. So that means the vertical extent of the water will be from 4 to 8. And keep in mind these are These are slices that are being made horizontally if you look at the cross section, but it's being we're looking at it from a vertical perspective. So from four, 28 and now to find a cross sectional area, well, it's a cylinder. But the cross section taking a horizontal slice of that of a cylinder is a circle, so we will use pi R squared. Define the cross sectional area and our radius is three three squared is nine. So a cross sectional area is nine. Hi, and now the distance the slices must be lifted. Well, let's see, It's we're being told that it we needed to be lifted two meters above the top of the cylinder cylinder itself is eight meters. That means it's gonna have to be lifted. 10 minus. Why? That's the distance that the slices must be lifted from a given point. And now we Plus, we have our formula right over here, and we're gonna plug in. Are all of our values into the variables in the equation? So we have from 4 to 8, um, P is gonna be 1000. That's usually the standard with water. Gravity is gonna be 9.81 We have are nine pie. And this could be this could all be brought out to the front country. Go about solving the integral. So this can all be consolidated to the front, um, or write it out 1000 times 9.8. And that's multiplied by nine. Hi. Let's find the integral of 10 minus y stuff. Be 10. Why minus why squared over two and that's from for 28 We'll take this a step further, and we'll plug in our values. When we consolidate the left part of or the Constance. We will get a value of right that Niedere 88 1002 190 hi. And that's gonna be multiplied by 80 because we're plugging in the eight minus 64 over to, and that can be rewritten as 32. And then that will be subtracted from what we get when we plug in four. So that will be 40 minus four square to 16 16. Divided by two is eight. And if we solve this out, will get a value upon footing this all into our calculator of approximately 44 33 415 And that's 0.6 Jules, and that is your answer for the first part. Now, for Part B, it's much of the same. The only differences were asked to find the work done toe lower to remove the lower half of the tank, so the only difference here would be the vertical extent of the water would be from zero 24 Everything else will stay the same. So let's plug that back into our equation into a value we get so same exact set up Onley different 0 to 4. Um, this will go a bit faster now, and that's multiplied still by nine pie, do you? Why? And let's plug in the values we know. Um, so we have bring this 1000 that we plug in for the pressure. Multiply by nine 0.81 multiple that by nine pie. And then we could find the integral of 10 minus y, which you'll get 10 y minus y squared over two. And that's on a boundary of 0 to 4. Just to check what we got above, we got 88,290 pie. So 88,000, 290 pie, and now it's plug in our four, and we're gonna get 40 minus 16 over to the same thing as eight. So 40 minus eight is 32. And while most apply these two together and if we plug them into a calculator, you're going to get a value of 8866 eight 31 0.1 jewels and that's it should be an approximation sign, and you'll notice that this is exactly double of what we got for part A. And if you take a step back and think about it, this makes sense. If the waters further down, you're gonna have more pressure. So it's gonna take more force or more work to remove this water, pump out this water from the lower half the tank. So you answer for C is it takes two times more work for pork be, and that is how you do this problem.

This question were given a conical tank with the bass radius of 0.5 meters a height of two meters and were asked to find the work required to pump water to a height of one meter about the top of the tank. So for this type of problem, we can use the formula. Work equals two integral from A to B road G A. Of Why Dio who? Why do you Why so a of why is the cross sectional area of the tank and d of why is the distance that the water has to travel? So let's start with a of why we can see that this is a Cohen. So the cross sectional area would be a circle. So a of why it's just pie are square. Um, so I just write that down first, but are we want to represent are using got a variable? Why so first we want Teoh have ah cornet system for our model. So let's just use it next and why. And let's say the very bottom is where zero is, since that would make the most sense for before this type of problem and we know that this point would be. Is there a 0.0.5 Comma two. We are trying to get the water to hide off three in this case because that's one meter above the top of the tank. And of course I'm the bottom, which is 00 Um, so we have X and y. We can see that the radius is just I just put it in red. The radius is just from here to there, which it would be the X value on. That's nice, but we needed to We need our area to be in terms of Why so first we might want to find out the equation of this line just dying in the line here and then solve it for X so that we can have ah, function in terms of why. So let's do that so we can see that the slope is why. To minus why one so to minus zero. Divide by X two months, x one sos 0.550 So that's two over is opened five, which is four and we can use the point slope form which used a top point. So it will be a Y minus two. You cause four x my 0.5, which you can see. It's just four x minus. So what are X minus two? And of course, the minus two. Castle also was just Why equals four X. In other words, X equals went over four times. What? So that's what art, since the radius excess the radius. So we pi times one over for why square all squared. And then I'll put that in later. Uh, I want to figure out what the of why is first. So we know that if I look at the bottom point, that point has to go up three all the way. Three units. Whereas the top point has a goal by one unit, and you can sort of see it. How is how these relates three minus zero Would be 33 minus two will be Give us that one. So it's always three minus whatever. Why value is that's what the d y d of why would be so d of why it is just three minus. Why so now we have basically everything except for A and B for off the Lauren upper bound. So we know that we're going from zero all the way to three. But for the foot, um, formula, this be represents just this part represents So is your circle. This this part represents force of gravity. So that's mass times gravity where masses roll times the cross sectional area. Um, sorry. This part together with D Y is, is the force of gravity. And how that works is that, um it only applies for what mass we have. In other words, it's Oh, it's only from 0 to 22 because this one extra height, it is how much we're lifting it by which is represented already by Thea. Why? So our work is just from integral of integral of 0 to 2 of Roe, which is 1000 for the mass density of water times 9.8 and then a of why would be high r squared which mean it's pie r squared would be won over four. Why square and d of why is three minus? Why do you Why So the 1/4 squared, I can just multiply that first we went over 16 and all the rest of these numbers are constants, which I can take out off the integral. So we're 1000 9.8. Hi. Over 16. And now I'm left worse. Well, I square three miles. Why do you? Why s so we can do a little bit of math here. So nine 90 9800 pi over 16 on into grow from 0 to 2. This will be three. Why square minus y que do you? Why, Um, which would be dying 800 high over 16. So the integral three y squared we know that integral y squared iss wife Courtney Why would be a y cube over three? So why Q over three times to readjust y que and integral. Why, Q would be a wide 4/4 and we are you guys going from 0 to 2? So that's 9800 high over 16. So, too So the second term, we're just zero because I can see when I penguins there were here and there the whole thing with just a zero. So to Cuba's eight. And to do before 16 over for So this used me. Well, 9800 high over 16 times. So eight minus four. Just for which gives me a value of 7696 for nine jewels, which she can around Teoh Sand 0.7 times, 10 to the three. That's it.


Similar Solved Questions

5 answers
Consider the Euler equation 212y" 3ry Use the method of Frobenius to solve this equation: That is look for solutions of the form and find A by solving quadratic equation: change of variable to In(z) transforms Euler equation into linear ODE with constant coeflicients_ Thus write yls) w(u) and find such ODE for w(u): solve for the general solution w(u): then find y(s) = w(ln(z)) You should reproduce the answer in (A):
Consider the Euler equation 212y" 3ry Use the method of Frobenius to solve this equation: That is look for solutions of the form and find A by solving quadratic equation: change of variable to In(z) transforms Euler equation into linear ODE with constant coeflicients_ Thus write yls) w(u) and f...
5 answers
SupposeJlsY)=3/_ +2Ky - Sw_ +4yFind all critical points0(0, 0) , (0.9475,-1.2849) , and (0.6192,0.6849)b) @(0.9475, 2849) and (0.6192 ,-0.6849)0(0,0). (0.9475. 2849) , and (0.6192 -0.6849) (0,0), (6.3597,-2.5698) , and (4.1070, 1.3698)(0,0) , (63597,2.5698), and (1.1070, 3698)None of the above_
Suppose JlsY)=3/_ +2Ky - Sw_ +4y Find all critical points 0(0, 0) , (0.9475,-1.2849) , and (0.6192,0.6849) b) @(0.9475, 2849) and (0.6192 ,-0.6849) 0(0,0). (0.9475. 2849) , and (0.6192 -0.6849) (0,0), (6.3597,-2.5698) , and (4.1070, 1.3698) (0,0) , (63597,2.5698), and (1.1070, 3698) None of the abov...
5 answers
(d) The 3 Graces were carrying baskets of apples, and in each was the same number_ The 9 Muses met them and asked each for apples and they gave the same number to each Muse and the 9 and the 3 each had the same number. Tell me how many they gave and how they all had the same number. (This problem is indeterminate. Find the smallest permissible solution.)
(d) The 3 Graces were carrying baskets of apples, and in each was the same number_ The 9 Muses met them and asked each for apples and they gave the same number to each Muse and the 9 and the 3 each had the same number. Tell me how many they gave and how they all had the same number. (This problem is...
5 answers
Find the first and second derivatives of the function f(x) = (x2 + 4)5f"(x)f"(x)
Find the first and second derivatives of the function f(x) = (x2 + 4)5 f"(x) f"(x)...
5 answers
0 Ginvera K# all QUESTION 6 2 soiutiongxforsy = 0
0 Ginvera K# all QUESTION 6 2 soiutiongxforsy = 0...
4 answers
The perimeter of an equilateral triangle with side length $ is 3s. Leslie is using wooden equllateral triangle with 15-inch sides as a stand for her newest sculpture: What is the perimeter of the stand?0 36mJ2f1) 9032 9104 bobssh 310qqu2 I9vI (srt Jesvab abn oingbngq 29v [ [ aea [Jtob 6e36 2ie [1j w29[109 IJo uoripuo dj 2 [qopg 910i gv9 tinumo? MIUJ [LiW nofjev [62 adT 0;
The perimeter of an equilateral triangle with side length $ is 3s. Leslie is using wooden equllateral triangle with 15-inch sides as a stand for her newest sculpture: What is the perimeter of the stand? 0 36mJ2f1) 9032 9104 bobssh 310qqu2 I9vI (srt Jesvab abn oingbngq 29v [ [ aea [Jtob 6e36 2ie [1j...
5 answers
Question 6 points) Determinez sE la serie suivante converge absolument, conditionnellement ou diverge: Eei"_
Question 6 points) Determinez sE la serie suivante converge absolument, conditionnellement ou diverge: Eei"_...
5 answers
Find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.$$8 y^{2}-2 x^{2}=16$$
Find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices. $$8 y^{2}-2 x^{2}=16$$...
5 answers
50 Ji 2 RZ6e' +0)None of these3 (e'
50 Ji 2 R Z6e' +0) None of these 3 (e'...
5 answers
VI: Retrosynthesis (10 pts) Determine retrosynthetic pathway t0 torm the following target molecule Also draw the synthesis in the fonward direction With reagents draw full mechanisms_ donofIPrPnRetrosynthetic Pathway:Fonward Dlroction:
VI: Retrosynthesis (10 pts) Determine retrosynthetic pathway t0 torm the following target molecule Also draw the synthesis in the fonward direction With reagents draw full mechanisms_ donof IPr Pn Retrosynthetic Pathway: Fonward Dlroction:...
5 answers
Find the number of nonisomorphic simple graphs with seven vertices in which each vertex has degree two.
Find the number of nonisomorphic simple graphs with seven vertices in which each vertex has degree two....
5 answers
Use the method of maximum parsimony to estimate the most likelyancestral trait (square, circle or star) and and how many changeswould be needed to explain the tree shown, Please explain answer; it would really help me study for myfinal! Thanks!!
Use the method of maximum parsimony to estimate the most likely ancestral trait (square, circle or star) and and how many changes would be needed to explain the tree shown, Please explain answer; it would really help me study for my final! Thanks!!...
5 answers
5.0 [0/1 Points]DETAILSPREVIOUS ANSVVERSDCALCLT0Use a double integral to find the area of the region. The region enclosed by the curve r = 5 + 2 cos(0).Need Help?Read It
5.0 [0/1 Points] DETAILS PREVIOUS ANSVVERS DCALCLT0 Use a double integral to find the area of the region. The region enclosed by the curve r = 5 + 2 cos(0). Need Help? Read It...
5 answers
20 Aiven 4Z (USx Tind X 44
20 Aiven 4Z (USx Tind X 44...

-- 0.019515--