First you’d think they should be at 5″ and 15″, but that’s not right because the tank is round, which means the bottom and tops have less volume per inch of height.\u00a0\u00a0 Then you think this is a problem is an integration problem – which it can be – but the integration becomes extremely hairy.\u00a0 Then, you find you can back up and use a geometric method (and when you can’t reduce anymore) use a numerical method to solve it.\u00a0 So let’s get started!<\/p>\n
We see that needing the actual volume of the cylinder is unimportant since you can solve this problem with just the cross-section – which is a circle.\u00a0 What you want is a circle with a chord across it in which the volume between the chord and the outer wall is 1\/4 the capacity of the circle.\u00a0 So, you draw a diagram, and get started!<\/p>\n
Unfortunately, you see that the equation becomes very difficult to solve analytically – and one must resort to numerical methods <\/a>to get an actual solution.\u00a0 I used the Mathematica online site, but you could easily use the Newton-Raphson<\/a> method as well.\u00a0 Whatever way you use, you find that he should mark the 1\/4 tank line 5.96027 inches from the bottom of the stick.\u00a0 3\/4 and 1\/8th values are also shown.<\/p>\n The value of this equation can quickly be used to calculate 1\/8, 1\/16, and all other desired fill marks by simply changing the 1\/4 * pi * r^2 line whichever fraction you’d like. In fact, you can graph it to get any value: Ignoring negative volumes, you see that the tank’s volume compared to it’s theta (roughly equivalent to height) forms a S curve, so that you can see that the height changes more rapidly w.r.t. volume when close to full\/empty than in the middle – just like we’d expect.<\/p>\n So, that’s your answer.\u00a0 Turns out, others have solved this since it’s a common problem with all kinds of other tanks (fuel oil, gas stations, etc).\u00a0 Here<\/a> and here<\/a> are other solutions that verify the same process and confirm that the final equation is unsolvable analytically.<\/p>\n Another person pointed out that most semi’s have TWO tanks – one on each side – which are connected by a balancing flow connector.\u00a0 So both tanks fill and empty evenly.\u00a0 Even though this seems to mess up the problem, it actually does not.\u00a0 In order to represent that situation, you simply multiply both sides by two (two tanks, two times the target volume) – which cancel each other out.\u00a0 You could have ANY number of tanks connected like this and the answer is the same.<\/p>\n It also doesn’t matter how long the tank is either (so long as the tanks are the same size if you have more than one).\u00a0 Finally, the theta angle you calculate doesn’t even depend on what radius <\/em>of the tank!\u00a0 So if you calculate the thetas for all the fill points, then you can calculate the 1\/4 mark on ANY size tank.\u00a0 Pretty nifty huh.<\/p>\n","protected":false},"excerpt":{"rendered":" The guys from Car Talk have weekly puzzlers, but this question wasn’t a puzzler, but this came from a truck driver who called in.\u00a0 He said (basically): “I have big cylindrical tanks on my truck that lays sideways under the foot step.\u00a0 Problem is that my gas gauge is broken.\u00a0 I have a stick that I can put in vertically, so if the gas is at the 20″ mark on the stick, it’s full.\u00a0 If the gas reaches the 10″…<\/p>\n![](\"https:\/\/i0.wp.com\/mattfife.net\/special\/tank-solution.png\")
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