I think this is the most elegant solution. Nice.
This is also a very nice solution using an analogous situation. I would have liked to seen it actually tried.
I asked my 18-year son, Spencer, who is a Material Science student “What is the height at which a horizontal lying cylinder is 1/4th full?” and he responded with 29.8% without having listed to the radio segment or being given any other information. He said he used integration by parts. Wonder if he’s right?
Yep, that gives the 5.96 inches answer that many have come up with, using different methods.
When you get older, you’ll learn that running a diesel engine completely out of fuel is a bad idea. And even when you get older, I don’t think you’ll be able to push that truck to the nearest fuel station after you run it out of fuel, so that you can partially fill the tank and try it again!
Well, this was fun! 
Ray was right, it’s a simple problem of integral calculus. Since the circle is symmetrical,and we are just looking for the ratio of 1/4 area vs whole area, we can consider just the 1/2 circle. This make numerical integration easier since we can then consider the explict function y = sqrt(r-x^2). Using a numerical integration package (VisSim) we can find the area under the curve and detect x when integrated area = 3/4 total area (pi*r^2/2). As the attached screen shot shows, for a 20" diameter tank 1/4 full=5.96", 1/8 full=3.65", 1/16 full=2.26"
For those who want to play with it, the diagram can be downloaded here:
www.vissim.com/downloads/software/cylindrical-fuel-tank-dipstick-check.vsm
A free version of the software to run it can be downloaded here:
www.vissim.com/content/free_vissim_viewer_download_request_form
A quarter of a tank will be indicated by 5.96 inches on the dipstick.
No calculus was needed.
This is a plain trigonmetry solution.
See the attached pdf file or enter this text into a programmable calculator:
((asin((R-A)/R)-90)/-180*(pi R^2)-(R-A) * sqrRoot(R^2-(R-A)^2))/(pi R^2) = fuel
A = length of fuel on dipstick.
R = radius.
Formula returns a value between 0 and 100% full.
Finally somebody got it rigth and using the rigth formulas, you can do all the filling, draining, driving and mileage count, but at the end you will waste a lot of time and money
I also did the math and came up with 5.96" or you can round it up to 6" and it only took me about an hour, and only because i keeped using the angle value in degrees, when i should have used radians.
My respect to you too, but i don,t think you can switch from one tank to the other, in any case if you use one tank or the other or bouth its always going to be the same, since bouth tanks have same capacity and shape.
The problem I have with finding a truck that indicates a tank at 1/4 full and then measuring it can be demonstrated by something I am sure we all have noticed, the gague moves slower when the tank is full and faster as you get near empty. This translates into a 1/4 reading whose related quantity when multiplied by 4 may or may not equal a full tank.
I think we have to do the math here.
I did a little research on integral calculus but i felt that it was easier for me to use the circular segment formulas and it goes something like this: A = R^2/2x(radians - sin deg.), so the tank is 20" diameter the radio is 10" we find the area first A = 10^2 X pi = 314.15… one quart of that is 78.53, now we change the circular segment formula to find the heigth, but i still needed the angle value so i did gess a few and put it in the formula, until i got an area that matched the area that we have (78.53), at the end the angle was 132.5 or 132 deg and 30 minutes, then we find the heigth: H = R (1-cos(deg/2)) = 5.97" or 6", i hope this is what you’re looking for, love your show.
what??? usable fuel??? unusable fuel??? were talkin about trucks here no airplanes
Go to www.wolframalpha.com, type in
pi/8 = integral sqrt(1 - x*x)
The answer is 0.404, measuring from the center of a circle with unit radius. So the distance from the bottom of the tank is 5.96, which corroborates the result that others have obtained with the geometric method.
Well, yes he should get his gauge fixed when he can.
All of the methods which employ driving and measuring residual are flawed because they assume level consumption under all conditions; besides they are mooted by the impracticality of needing to know when to stop and measure and the excess fuel consumption entailed in doing that.
Rich said his fill cap is on top and his dowel goes in vertically.
I would guess the purpose of wanting to know the 1/4 mark is to be able to anticipate refueling before reaching it. A good approximation is good enough.
It’s not that hard to integrate the area between a circle and a horizontal chord using integration formulas or mathematica online and JohnFID’s integral is correct. However, it can put put into a more useful form by dividing out the area of the circle (pi*a^2) to give the area fraction and then multiplying by the capacity, C, of the tank in gallons. This way, the length of the tank doesn’t matter. The resulting formula for the fuel level, F, in gallons is:
F = C*(1/2 + (arcsin(h/a-1) + (h/a-1)sqrt(h/a(2-h/a)))/pi)
If we graph the fuel level for a 200 gallon tank with a 10" radius, we get the attached plot. It is 1/4/ full when the level is 5.96" and yes, it would be quite a long tank.
I solved it using a spreadsheet “goal seek” function. I calculated the sector area, subtracted out the included triangle leaving the segment area which was targeted at 1/4 of the full tank. Using this method I got a depth of 5.960272 for 25%, and 14.039728 for 3/4. That’s close to 6 and 14 inches… Close enough!
Ivan
This is essentially the method I used using a spreadsheet. I solved the equations to calculate the segment area as a function of the included angle x. Then did a goal seek to find that angle when the segment area was at 1/4 the area of the circle; the height of the triangle was then easy to calculate. See my numbers posted below.
Ivan Awfulcoff
It doesn’t matter how many tanks are used, if they’re all used at the same time and the same dimensions, they’ll all be the same depth at 1/4 of the total fuel remaining.
were talkin about trucks here no [sic.] airplanes
That’s my point exactly! Everyone keeps addressing this problem like we are talking about a pick-up truck instead of a commercial truck. The fuel line in a commercial truck fuel tank doesn’t go all the way to the bottom of the tank. This allows for the sludge and dirt in barely-refined diesel fuel to collect at the bottom of the tank. Every once in a while, truck drivers have the remaining sludge at the bottom of the tank cleaned out, although now that diesel fuel has higher refining standards, that isn’t as often as it used to be. Since we are talking about a truck with a non-working fuel gauge, I am betting the truck is an older truck with more than 500,000 miles on the odometer. Therefore, if trucker Rich wants to know how much usable fuel he has remaining in both tanks, we need to know how much unusable fuel each tank holds.
The only point I am trying to make is that in order for trucker Rich to know how much farther he can go between fuel stops, we really should know the how far the fuel line ends in relation to the bottom of the tank. If you were trucker Rich, wouldn’t you want to know that?