2974: Storage Tanks
| Storage Tanks |
 Title text: We're considering installing a pressurization system to keep the tanks at constant pressure solely to deter them. |
Explanation
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This explanation may be incomplete or incorrect: Created by SUBSTITUTE CALCULUS TEACHER MR. MUNROE - PlsplsplsplsPls help . Do NOT delete this tag too soon. If you can address this issue, please edit the page! Thanks. |
In this comic two security guards is on a scaffold overlooking an area with two large tanks. One of them is instructing the other that he should "watch for calculus teachers trying to drill holes in their bases." Sadly they are not alert at this moment and Miss Lenhart is seen running away with an electric drill after having drilled a hole near the bottom of the tank furthest from the guards.
A common question in introductory calculus courses asks how long it will take a tank to empty. Assuming the tank is shaped like a cylinder, as appears to be the case in this comic, the amount of fluid left in the tank will follow a quadratic decay, as the rate of pressure decrease will be proportional to the instantaneous pressure at any given moment. Variations of this question may consider more complicated tank geometries too, in which case the pressure at the base will not be simply proportional to the volume of fluid remaining in the tank.
The joke in the comic is that calculus teachers, at least of the Miss Lenhart type, will actually drill a hole in a storage tank to demonstrate this principle, to protect against which a factory might station a guard to keep a lookout for them.
If this was supposed to be a practical demonstration, it seems that Miss Lenhart has not taken her class with her. It would also be very difficult also sneaking them in. But maybe they can watch from afar. It is, however, not a particularly useful way of teaching the math behind it.
Sneaking into such a facility without permission and damaging their property is of course illegal and, if caught, will get you charged with trespassing and destruction of property. Mmost teachers would not consider this a proportionate risk. The leaking liquid may also pose a safety risk, as implied by the warning triangle on the tank (⚠).
The title text jokingly alludes to the fact that by maintaining a constant pressure in the tank, the rate of flow would itself become constant, which would simplify the problem greatly and therefore make it much less interesting, or useful, as an example in a calculus course, thus frustrating the teacher and maybe get them to look for another set of tanks to use. If the pressure applied was higher than expected, it could also result in a nasty shock to the teacher drilling into it, as the liquid rushes out surprisingly fast.
Transcript
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This transcript is incomplete. Please help editing it! Thanks. |
- [Two characters wearing helmets are standing on scaffolding next to two large tanks labeled "Tank #3" and "Tank #4", with the person on the left talking. Miss Lenhart has drilled a hole into the base of Tank #4 and caused the liquid inside to leak out of the tank, then she runs away with the drill in her hand.]
- Left person with helmet: As head of security, your primary task is to monitor the storage tanks and watch for calculus teachers trying to drill holes in their bases.
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The symmetry of the truss intrigues me. Struts that are diagonal across the faces of the cuboids is normal, but is it a real thing to also use the body diagonal? Never seen that IRL, not sure if it makes sense from the statics. --172.70.247.82 22:16, 19 August 2024 (UTC)
- Struts as shown provide some left-right stability, but not as effectively as struts across the face would. They also provide some redundant front-back stability with the struts running along the faces. 172.68.71.90 14:47, 21 August 2024 (UTC)
- I mentally modeled the flexibility modes, and it very much depends upon whether the verticals are solid (with their resistence to bending playing a big part alongside the incident horizontals'/diagonals' exactvmethod of attachment) or are just sections of rod between a suitable receiver-'node' at each junction.
- in particular, the strength of any one of the three 'boxes' (each level between adjacent horizontal cross-sectional perimeters) is somewhat less secure, as a single level can 'fold' sideways over each side's vertical-diagonal strut (along with the respective front/back horizontals, held 'square' by the internal cross-brscing). Only the continuation and linking with the other 'boxes' really guarantees any innate stability, and if each node is free-twisting then the likely first result of any failure is that the tower topples forwards and/or backwards as it folds up due to the unbraced facing and hindside quadrilaterals.
- But it does depend a lot upon the exact nature of the linkages (which can only be guessed at), and other failure-modes could involve node-slippage if they merely grip the cross-braces to the entirely top-to-bottom poles and there's potential for sliding there instead of primarily rotation (or over-stressed failure in any given length of rod).
- The support (or additional pressure) provided by the access staircase is also probably a factor. It could even be the most important bit in holding it up! ...if firmly anchored at the other end and robust enough in itself.
- I'd definitely add other diagonals (including opposite-type body-diagonals, perhaps tied to the existing one as they pass right across each other), just to be sure. The more the better, of course, but there's probably a limit through diminishing concerns. And too many diagonals primarily in a helical pattern could concentrate forces into a particular type of rotational failure if you also add too much brace-weight in doing so. 172.70.163.120 15:07, 21 August 2024 (UTC)
Seems like a pretty menial job for the "head of security". I think he would delegate this to a security guard. Barmar (talk) 00:47, 20 August 2024 (UTC)
- They may be head of a department of one.172.70.85.139 08:50, 20 August 2024 (UTC)
The explanation mentions there might be more complex calculus examples where the shape might not be a cylinder. I think some further explanation could be added that this does not change the pressure (hydrostatic paradox) but indeed change the rate of emptying the object. If differing cross sections are relevant at all. 108.162.221.103 05:40, 20 August 2024 (UTC)
- Non-prismatic geometries are I think the ones being alluded to here, i.e a frustrum with the pointy end down will have a greater reduction in pressure for a given volume of flow towards the end than at the start, which may offset the reduction in absolute pressure. I've also seen examples where the flow rate is considered constant and the problem is to work out the fluid depth as a function of time, e.g. filling a pyramidal pool from a hose. 172.70.58.4 16:44, 20 August 2024 (UTC)
Its the most difficult job in history, even the best workers couldn't stand 1 day as head of security.I HAVE NO NAME (talk) 05:55, 20 August 2024 (UTC)
I have to admit, I thought I knew calc as I had two semesters of it, but I had to look up what he meant by this. Ouch 172.70.242.55 13:01, 20 August 2024 (UTC)student
If anyone could suggest something I can do for my class now that I can no longer drill holes in tanks, I'd appreciate the advice, thanks. Fephisto (talk) 16:18, 20 August 2024 (UTC)
Someone should do the math on the calculus problem as presented, as well as the algebra version. Laser813 (talk) 17:33, 20 August 2024 (UTC)
- Randall, like all good mathematics textbook authors, left the problem as an exercise for the reader. Does this happen often enough to warrant a tag? Paddles (talk) 05:57, 21 August 2024 (UTC)
- Yes... Transgalactic (talk) 10:10, 21 August 2024 (UTC)
Is there anyone else who thought the calculus teacher was abusing the tank as a model for the complex plane, demonstrating how to remove a singularity from a holomorphic function by puncturing the plane? I wasn't confronted with that particular tank-emptying problem in high school, so my first encounter with "holes" in maths was in complex analysis. The title text was a mystery. Transgalactic (talk) 10:10, 21 August 2024 (UTC)
As a mathematician, I'm surprised I didn't know about this idea. (It's definitely not my field!) I actually thought the flow would be constant, an algebraic problem. Oh, I'm sure I saw these types of problems in Calculus (and I remember problems like this in Differential Equations), but I thought those were just to make the math more complicated, not based in reality... So is it the weight of the liquid remaining above the hole that is the source of the pressure (i.e., would it be the same if the top of the tank were open), or is it the air pressure in the tank as the volume of liquid decreases and volume of air increases? Mathmannix (talk) 11:08, 21 August 2024 (UTC)
- A sealed-top would change the dynamics (like trying to pour the contents of a 2 litre (or whatever the US equivalent is) pop/soda bottle, it will tend to 'glug glug glug' intermittently unless you: a) incline the bottle to allow an optimum amount lf free akr back into the emltying bottle or, b) initialise the up-ended emptying with a spin sufficient to create a 'waterspout' effect up through which the replacement air can (more) freely pass.
- Though there are other possible factors, in that example, including the potential pressure of any self-releasing carbonation pressure (e.g. giving the bottle a shake, or a foreign body, before releasing the 'pour') and/or squeezing/'milking' the soft plastic container strategically to create another form of pressurised expulsion.
- For the 'classical' problem, one should probably assume sufficient inward venting (either an open/part-open top or a second hole drilled near the top to effect this purpose) as well as a reasonably unexotic liquid (neither molasses, cornstarch-mixture, anything that is actually a very fine dry powder, anything that reacts significantly with/upon air, any liquid very close to its vapour-point nor specifically supercooled helium) or any additional elements (stirrers, baffles, spongey inners, inner membranes or the contents being a layered combination of imiscable liquids of different densities that may or may not react slightly all across the interface plane). Most things that aren't actually exotic (and even a few that are, and might warrant a warning /¡\) are close enough to water to treat as if just that, at least under the further assumption that we're working at or around Standard Temperature and Pressure. But a slightly different density, viscosity and surface tension (plus the nature of the container, e.g. extreme hydrophilic or hydrophobic inner coatings where water is involved) could (in combination) drastically change the actual outcome given enough of the right kind of simultaneous differences imparted. 172.70.86.37 14:28, 21 August 2024 (UTC)
- We have these at my work: https://shop.snydernet.com/images/snyder-6370721n95402.pdf The viscosity is a huge deal, unless you can afford to waste a bunch of supply. 162.158.186.248 08:29, 23 August 2024 (UTC)
In order to remove calculus from the scenario one needs a pressurization system that keeps Pair + Pliquid, constant at the hole. This requires sensing the height of the liquid surface and increasing the air pressure as the surface drops, the relationship depending on the density of the liquid. Mjackson (talk) 22:02, 22 August 2024 (UTC)
I wish my maths teachers back in the day had come up with any examples like this and I might have got a notion that calculus can be good for something after all, that there is an actual justification of learning it. I still might have failed in grasping it, mind you, but at least I'd have the decency to feel just a tiny little bit bad about it. I realise this is probably making me the odd one out here, but I felt I should mention it because it's linked to a fundamental problem of teaching - teaching anything at all, not just maths: most of the time you need to sell to your students the notion that knowledge and skills outside of your fields of interests, those that certainly don't relate to any career you have in mind for yourself, are valuable enough to justify the pain of learning them. You see, generations of "dumb", "lazy" students like I used to be myself have got it right on instinct: anything complex and abstract is literally painful to learn. Thinking hard in fact generally hurts, even if you pretend to yourself you like it (as I do, too, but what we really like are the results we come up with when we're done). Psychologists recently hammered this fact home in a large meta study [1] [2]). So with something as complex and abstract as calculus, you'll have to sell to them it's worth learning anyway, even despite a very real prospect of being able to live a perfectly happy and successful life without it, and if it's only of the weak basis of "it may still turn out to come in handy at some point". I'm a teacher myself, in case you're wondering, and if I taught maths and calculus and suchlike, transferring to it the same "signature" hands-on approach I'm somewhat known for using in geography, it'd be quite possible you'd see me drill an actual hole into an actual barrel (and likely make a show of installing a tap to make it reusable, too). PaulEberhardt (talk) 10:21, 24 August 2024 (UTC)
If the tank were sufficiently pressurized, they might end up having to mop up careless calculus teachers. The Hydraulic Press Channel has a series of videos showing the results of fittings popping off of over-pressurized tanks. It did not bode well for any ballistic gel dummy heads that got in the way. These Are Not The Comments You Are Looking For (talk) 03:17, 26 August 2024 (UTC)
- Don't worry. Us teachers are professionals, because we get paid for it. 😁👍 PaulEberhardt (talk) 07:38, 29 August 2024 (UTC)
