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Numerade Educator

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Problem 12 Medium Difficulty

A milk truck carries milk with density $ 64.6 lb/ft^3 $ in a horizontal cylindrical tank with diameter 6 ft.
(a) Find the force exerted by the milk on one end of the tank when the tank is full.
(b) What if the tank is half full?

Answer

(a) Force exerted by milk on the tank $\approx 5479.57$
(b) Force $=1162.8 \mathrm{lb}$

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Video Transcript

in this problem we're given That truck carries milk in a container with a circular cross section and whereas to find the total force acting on that container one D containers filled with milk so the density weight density off the milk is given to be £64.6 per feed. Cute. We know not the radius or total diameter X six feet. So it means that date radius is three b, not a total force is equal to force for. We know that force is equal to pressure times area, and that is equal to row tun g times deep pep that the pressure is measured multiplied by the area. We know what you know. What g is. We can, um depending on the origin. Me. How we can find another thing is to find the area. Now, since we're interested in pressure or the force acting on this cross section, view it like to find the area off the circle cross section. Now, the thing is, don't depending on the that smart side. And this is the depth depending on that, uh, the with off this object changes since with off the subject changes, we could write the area off distance ship where the height d y as omega or w y times d want. And we would like to find an expression for this. W what now we know that we have a circular cross section, So let's drop it again. Let's assume the center of this circle to be the origin. So 00 Let's say that ex starts from here. Why starts for there? And, um, we know that then the question of a circle is excreted. Plus y squared is equal to the radius square, so that is three scraped from the species. That access than equal to skirt it up nine minus for scrap. Now does as your radius on one end is a function of what? So that would be skirted off nine months west card on one side, nine minus y scarred on undersized total. What would be then? Um so from this we see that w why wouldn't be two times skirted off nine minus y squared and area would not be what multiply but do you want? So that would be two times carted up nine minus y squared times d y. This is the area we know what it is. We know what he's less when an expression for that Now, since we assume the center of this circle to be the origin and since we know that the total radius so the origin to the upper surface is three. And since the depth is measured from the operas surface, the idea is that if we go a CZ, we got the print. The pressure will increase. Adept as a functional. Why will lend me three months? Weiss ordered Total radius will still be three sort of total force will then be integral row g times that is three minus y. An area is too skirted off. Nine minus y scribed. Dean, Why all right now, innit? It's German. Be a limits off this inter girl. This is our origin. So this is Michael's your level. And since the rate is history, this would be Michael's negative three level, and this upper would be Weicker's three level. And this is the total height off this object. So the limbs often in jungle, would be negative 3 to 3. Now we need to calculate this integral. Let's use this remaining space. We can we know that road G and two or constants. As a matter of fact, we were given the, um, weight density so we can write this one as to row G Inter goal from negative 3 to 3. Behalf three times skirted off nine minus y Scribd. Do I minus integral from negative 3 to 3 white time skirted off nine months. Why skirt? D'oh was closed. The break it. All right, so the Inter go second Inter go as or the argument off the central is an auto function, it means that this integral will be zero. So we need to do is to let the first interview forced on me. Since three just constant six times road times G integral from negative 3 to 3 skirted up nine months west guarded by Now we know that this is an expression for X, so this integral, basically easy area off health circle. So this integral is actually the area office. 1/2 off this circle sort of force would be six lamb short time G pie. Since radios is 23 squared or two were given row G 64.6. So that will be six times 64.6 times five times three squared over two and we don't find the total force to be 5479.5 £57. All right, in part B, we have the same exact German t. Understand this container or half off this container is filled with fluid. We're going to use the same exact procedure. We know that X is a functional for as nine months y squared. You know that the area off a 10 strip just like part a is two times skirted off nine months once cried times d y now the only difference is less drawing a circle again. We know that this is our origin and we know that health is filled with milk. So the reference for the pressure will be this step right. And this will be then if this r y and if this is our ex, the depth will be done. Negative one. So the total force as p times saying that'll be done raw times, g times, negative y times the area two times skirted of nine minus y squared times d y. And to find total force, we would need to integrate it. What are the in tradition limits. Well, we know that here is our Y zero level, and this is what is it with the negative three level sort of limits? Will that be negative? 32 here. All right. So the force would then be integral. Negative 3 to 0 or force would then be less right to constant outside to intrigue. Also to 10 0 times G internal negative, 3 to 0. We have negative. Why multiplied by skirted up nine months. We described D Why now? What Say that? Uh, we can write this one, actually, as since we have negative here. And since limits are negative 3 to 0, we can write this one as positive to row G. You tackle from 003 Why? Skirted up nine minus Weiss kind do you want Now we're going to say that nine minus y skirt is our new variable. That is you. So the limits off you would be when y zero u would be nine. And when wise three, you zero. If this is the case negative too. Why d y would be d you sort out why d y would be equal to negative d'you or to soar new interval is then f physical too negative to Rogie Integral from 9 to 0. Negative skirted off you. D you or two from this? We see that then, uh, these will cancel out. So we have Rocchi. What negative will make positive Rocchi off to you three over to over 23 Weren't you changes between nine and zero. From this? We see that F is that an 18 plans Row G and were given Rogie at 64.4 said I will be 18 times 64.4. So did force one. That is what half of it is felt with milk as 1000 162.8 perhaps?