2974: Storage Tanks

2024-08-21T20:25:29Z

LimitOfN: Add the possibility of multiple holes being drilled as possible explanation for the amount of water

{{comic
| number = 2974
| date = August 19, 2024
| title = Storage Tanks
| image = storage_tanks_2x.png
| imagesize = 321x251px
| noexpand = true
| titletext = We're considering installing a pressurization system to keep the tanks at constant pressure solely to deter them.
}}

==Explanation==
{{incomplete|Created by SUBSTITUTE CALCULUS TEACHER MR. MUNROE ... WITH THE POWER DRILL ... IN THE CONSERVATORY - Do NOT delete this tag too soon.}}
Two construction workers in hardhats are on a scaffold, overlooking an industrial area with two large tanks. One is instructing the head of security that "your primary task" is to "watch for calculus teachers trying to drill holes in their bases." [[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 {{w|calculus}} courses asks [https://www.haywardflowcontrol.com/media/contentmanager/content//downloads//VessTime.pdf how long it will take a tank to empty.] Assuming the tank is shaped like a cylinder and is open to the environment (so that air can enter the tank to replace water as it flows out), the amount of fluid left in the tank above the hole will follow a quadratic decay, as the rate of pressure decrease will be proportional to the instantaneous pressure at any given moment. (Advanced variations of this question may consider more complicated tank geometries, in which case the pressure at the base will not be simply proportional to the volume of fluid remaining in the tank.) Specifically, the rate at which water flows out of the hole is governed by {{w|Torricelli's law}}, which states that the speed 𝑣 of the water flowing out is proportional to the square root of the height ℎ(𝑡) of the water above the hole.

Also, it's common practice for STEM teachers to use real-world examples to make abstract concepts more memorable for their students (though practical demonstrations like this wouldn't generally be used to teach calculus more than once per term per class).

This comic combines these ideas, implying that there's a rash of calculus teachers that go around drilling actual holes in storage tanks to demonstrate the underlying math principle, so much so that the primary job of the head of security is to prevent this pedagogically-motivated destruction. Perhaps Miss Lenhart's class is watching from afar. (Adding to the drama is the warning triangle (⚠) on the tank, implying the liquid is toxic.)

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 from a calculus exercise to a simpler algebraic one, thus deterring calculus teachers — though it may inadvertently attract similarly adventurous algebra teachers. (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.)

An easier deterrence option — compared to installing a pressurization system — would be to make the problem more complex by simply having the tanks fully closed to the environment, forming a vacuum as liquid drained out, which would represent a more complicated problem suitable to an advanced undergraduate or graduate course in Fluid Dynamics or Advanced Differential Equations.

===Analysis and Calculation===
Observation of the comic suggests the following assumptions:
* Tank height above hole: ~20 feet
* Tank radius: ~6 feet
* Type of drill bit: Normal (not a hole saw)
* Size of drill bit: 1 inch (largest commonly available twist drill bit)
* Goal of Miss Lenhart: To demonstrate quadratic decay to her students

With a tank that is 20 feet high, has a 6-foot radius, and a 1-inch diameter drill hole, it would take approximately 21.5 hours for the tank to empty completely — too long for a suitable class demonstration, and likely to be fixed by nearby workers who notice the leak.

That suggests the exploits are being filmed and shown later in an abridged (time-lapse) form.

Another way to put this is that her drill and drill bit seem to be laughably small for the job at hand.

(If she is using a large "hole saw" drill bit — which can be up to 6 inches in diameter — then that would explain the quantity of gushing liquid, and the tank could be emptied in about 36 minutes, but the drill doesn't seem to have that kind of bit on it.)

The most likely explanation to all this is that Randall didn't think through the drill and drill bit size in relation to the apparent hole size, leaving it only to nitpicky editors of a comic explainer website to even notice and care.

Alternatively, the teacher could have drilled multiples holes.

==Transcript==
:[Two guards with helmets are standing on scaffolding to the left of two large tanks with labels at the top. The tanks are cylindrical with a smaller base than the tank above it. The left tank has a small sign with unreadable text and near the bottom of the right tank there is a triangular warning sign with an exclamation mark inside it and a line of unreadable text below it. The guard on the left is talking to the other guard. Miss Lenhart is seen running away from the right tank with an electric drill in one hand. There is a hole in the base of the right tank which has caused the liquid inside to leak out of the tank splashing on the ground in the direction of Miss Lenhart.]
:Left guard: 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.
:Label: Tank #3
:Label: Tank #4

{{comic discussion}}

[[Category:Characters with hats]]
[[Category:Comics featuring Miss Lenhart]]
[[Category:Math]]
[[Category:Physics]]

explain xkcd - User contributions [en]

2974: Storage Tanks