Hello! I am a recent graduate from Hamilton College (BA in Physics) and soon to start my PhD at Wesleyan University. Last summer I submitted a manim video to SoME1, then-titled "The Physics of Light and Rainbows". Grant reviewed it and encouraged me to make more videos (he also suggested a better title, which it now has).
This particular problem has been on my mind for a while, but I'm not sure how to present it in a novel and interesting way. I'm looking for someone who can help me tell the "story" of this problem.
Quick Summary
A trucker once called in to an episode of car talk, explaining that his fuel gauge had busted and he needed to use a dip-stick to determine whether the tank was at least 1/4 full. Since the cylinder is on its side, we're really asking "at what height above the bottom of a circle do you position a horizontal line, such that the area below the horizontal line is 1/4 the area of the total circle?"
Visual aid from the article linked below.
I think this is a really interesting problem with a really nice physical motivation, and I've seen a few articles online answering the question (such as this one). However, every answer I've seen in my googling has only gone so far as to set up the integral, and then give a numerical answer, rather than an analytical one. I'd like to arrive at an analytical solution.
Here's the problem: there isn't one. The final step of the integral is a transcendental equation, and as far as I can tell, there is no closed-form analytic solution. That's the novel aspect I want to delve into.
Target medium
In a perfect world, I would like to make a video with audio snippets of the original episode, to make the viewer feel like they're part of the discussion. Since that's probably not fair use, my backup plan is to translate those "audio snippets" to text and deliver a blog-post-style explanation.
I'm also assuming my audience has taken an integral calculus class, even if they don't remember everything from that class. They just need to know what an integral is and why we would set one up to solve an "area under the curve" question.
More details
My ideal storyboard is this: even if it's the case that copyright law prohibits us from sharing insights from the episode in any capacity, I'd like to start by insights from the original discussion. I want the audience to feel like they're part of the team solving this problem.
It's hard to get started without a visual aid, so here's something I threw together in Desmos:
Our problem: for what height is the cylinder 1/4 full?
The first insight that the trucker offers is that the position of the vertical line is "probably not 1/4 of the way up", and the brothers agree. I want to go into why that is, briefly, because I think that may be most people's first guess.
Next, the brothers realize that this is an area problem, not a volume problem, and that "you can do this in two dimensions". This is clear from the visual aid, but the brothers didn't have one, so this is another important insight. And at this point, it is abundantly clear that we need to compute the area under a curve, so we know for a fact this is an integral calculus problem.
I worked through the integral in this Reddit post and found that the last line was transcendental. I want the bulk of the video/blog post to be going through this integral, and getting to the final line.
Once we have that last line, I'd like to point out two things:
There are many more practical ways of solving this problem. The brothers on Car Talk advised the caller to stop at a popular truck stop, ask if anyone had a quarter of a tank left, measure using the dip-stick and mark it there. (Emphasize that this is probably what you or I would do if this was your real-life situation).
By setting up the integral, we have enough information to solve this numerically, which is also kinda fun to do:
Short python script for numerically finding the height of the line.
The reason I've sat on this for a while is cause it feels like a weird story to tell. We try working for a result and we hit a wall. I want to go further by proving, rigorously, that this has no analytical closed-form solution, but truthfully I don't know if there is a way. It just seems morally true that it doesn't. In the absence of that final motivator, that oomph to get us through to the finish line, I want to conclude with a quote from Russian mathematician Andrey Kolmogorov:
"The mathematicians always want that their mathematics should be pure, that is, strict and provable, wherever possible. However, the most interesting and realistic problems could not usually be solved in that manner. Therefore, it is very important that the mathematician should be able to find the approximative (not necessarily strict but effective) ways of solving such problems".
-A. Kolmogorov
Contact details
Reach me below in the GitHub comments, or at mhanrahan42@gmail.com if you want to collaborate!
About the author
Hello! I am a recent graduate from Hamilton College (BA in Physics) and soon to start my PhD at Wesleyan University. Last summer I submitted a manim video to SoME1, then-titled "The Physics of Light and Rainbows". Grant reviewed it and encouraged me to make more videos (he also suggested a better title, which it now has).
This particular problem has been on my mind for a while, but I'm not sure how to present it in a novel and interesting way. I'm looking for someone who can help me tell the "story" of this problem.
Quick Summary
A trucker once called in to an episode of car talk, explaining that his fuel gauge had busted and he needed to use a dip-stick to determine whether the tank was at least 1/4 full. Since the cylinder is on its side, we're really asking "at what height above the bottom of a circle do you position a horizontal line, such that the area below the horizontal line is 1/4 the area of the total circle?"
Visual aid from the article linked below.
I think this is a really interesting problem with a really nice physical motivation, and I've seen a few articles online answering the question (such as this one). However, every answer I've seen in my googling has only gone so far as to set up the integral, and then give a numerical answer, rather than an analytical one. I'd like to arrive at an analytical solution.
Here's the problem: there isn't one. The final step of the integral is a transcendental equation, and as far as I can tell, there is no closed-form analytic solution. That's the novel aspect I want to delve into.
Target medium
In a perfect world, I would like to make a video with audio snippets of the original episode, to make the viewer feel like they're part of the discussion. Since that's probably not fair use, my backup plan is to translate those "audio snippets" to text and deliver a blog-post-style explanation.
I'm also assuming my audience has taken an integral calculus class, even if they don't remember everything from that class. They just need to know what an integral is and why we would set one up to solve an "area under the curve" question.
More details
My ideal storyboard is this: even if it's the case that copyright law prohibits us from sharing insights from the episode in any capacity, I'd like to start by insights from the original discussion. I want the audience to feel like they're part of the team solving this problem.
It's hard to get started without a visual aid, so here's something I threw together in Desmos:
Our problem: for what height is the cylinder 1/4 full?
The first insight that the trucker offers is that the position of the vertical line is "probably not 1/4 of the way up", and the brothers agree. I want to go into why that is, briefly, because I think that may be most people's first guess.
Next, the brothers realize that this is an area problem, not a volume problem, and that "you can do this in two dimensions". This is clear from the visual aid, but the brothers didn't have one, so this is another important insight. And at this point, it is abundantly clear that we need to compute the area under a curve, so we know for a fact this is an integral calculus problem.
I worked through the integral in this Reddit post and found that the last line was transcendental. I want the bulk of the video/blog post to be going through this integral, and getting to the final line.
Once we have that last line, I'd like to point out two things:
Short python script for numerically finding the height of the line.
The reason I've sat on this for a while is cause it feels like a weird story to tell. We try working for a result and we hit a wall. I want to go further by proving, rigorously, that this has no analytical closed-form solution, but truthfully I don't know if there is a way. It just seems morally true that it doesn't. In the absence of that final motivator, that oomph to get us through to the finish line, I want to conclude with a quote from Russian mathematician Andrey Kolmogorov:
Contact details
Reach me below in the GitHub comments, or at mhanrahan42@gmail.com if you want to collaborate!
Additional sources pertinent to the discussion: