It’s not Calculus, it’s trig! The area of a unit circle below a chord subtending an arc of A radians is A / 2 - sin(A) * cos(A) for crying out loud! Of course, solving for A is a pain. I recommend linear interpolation, but it’s all pre-calc. Anyway, I feel better.
Followed the Geometry route to solving using the pizza slice with a triangular section removed. Results agree with the Pringle Can measurement. Attached a graph and formula.
God, I need a life.
How would you find the 1/4 full and 3/4 full locations on a tank that is egg shaped?
It seems as though the best solution will probably be the simplest solution.
An overly complex solution is probably not going to be the best solution for drivers(including truck drivers) in the long term.
For the folks who suggest running the tank dry, doing so with a diesel engine is usually a real pain. Re-priming an out of fuel diesel can be a real process.
Here is my two bits, V=pi*(r#2*(height of cylinder)
oops! I don’t have a “pi” key. #=pi
And friends, no one gets points for the correct spelling of mileage (milage.) Or for using diesel instead of gas. This should be fun!
this sounded familiar . . . . and it was!! Waaaaaaaaayyy back in 2002 the very same puzzle was presented with the same actor, Rich, from Florida who was on the road in Missouri!!
mathforum.org/library/drmath/view/61752.html
Bob
Looks like we’ve been had!
As I said on the other post for this issue:
As a truck mechanic who has also filled his fair share of these thing, remember:
-
The fill cap is not directly above the top of the tank but slightly off set. Otherwise, you are trying to shove that big nozzle into a hole under the cab. I’ve had to turn these tanks that worked loose and the cap was right under the cab and near impossible to fill.
-
Law requires the tank not be filled to more than 95% of liquid capacity. It’s stamped right on the tank next to the fill cap. This is to allow for heat expansion and to prevent overflow.
Also, yes the tanks are connected by a cross-over pipe. There is also a shut off valve on each tank so you can drain one at a time if one needs replaced. We ran into this on some brand new Freightliners that were built in Mexico. They installed one valve backwards and they literally ran off of one tank from Mexico to Idaho. LOTS of fuel stops.
- Go to your local bulk lubricant supplier. They have measuring sticks all ready marked.
I tried using a trigonometric formula but don’t remember enough trig. I was confused because the Angle from the center to 1/4 way points would be larger than 90 degrees and the sine tables don’t go larger than 90 degrees. Anyway, I drew a circle on a piece of graph paper, counted all the squares in the circle. I ignored squares that were not at least half a square. Divide by four, count that number of squares up from the bottom and you find (roughly) that a quarter tank would be about 5.83 inches from the bottom.
Sounds like there is a practical solution and a theoretical solution. The practical solution(i.e. fill the tank with 1/4 tank of diesel and then mark the dowel) may be the best for truckers to use, and the theoretical solution(trigonometric solution) may be best for people to use as a puzzle.
The practical solution takes into account such factors as rounded ends, internal baffles, offset fill neck, fill to 95% capacity, etc. and it can also be used on any shaped tank.
Very interesting discussion.
I used the pie and triange approach too. I got about 23.8 degrees for 5.96 in from the top. I only had to use sin and cos - my trig is rusty!
http://goo.gl/qzw9U
Drove me nuts, but calculus is not needed:
Reducing the problem to the area of an arc of the unit circle, the arc Area A = r^2/2(c - sin©). r=1, c =PI for the semicircle, which yields 1/2 (PI -0) = PI/2 ~= 1.576.
So half the area is PI/4 or _= .785
Let angle t be the number of radians such that 1/2(t - sin(t)) = PI/4 ->
t-sin(t)= PI/2.
From a trig table, we find that t ~= 2.31 radius.
Plugging this into a arc area calculator, we get an area for .785 arc height of 0.596. Scaling this up to a dipstick, the mark for 1/4 from empty should be 5.96 inches from the end of the stick.