There is one-quater of a tank left when the dip stick measures 7.008 inches of fuel, assuming that the tank is level, has flat-plate ends, and the dip stick is vertical. That is, for any diameter, the dip stick measures 35.04%.
Citation: Burington, Handbook of Mathematical Tables and Formulas, Section Elementary Geometry, at page 12 (R.S. Burington, 1933)
So many answers have shown the correct height (approx. 6 inches) ? some being calculated using geometry and trig, some using calculus, some using a scaled down representation of the tank (like a Pringles can), and some calculated using tank simulation software ? it?s utterly amazing that anyone would still offer up an answer to the contrary. If I draw the circle with your measurement, it looks like the attached file. The shaded area created is approx. 31.24% of the total area, not 25%. I don’t know what table or formula you’re looking at, but it’s not correct.
Then we?ve got somebody who purports to be an MIT graduate talking about the density of gasoline, which is irrelevant because 1) it doesn?t matter what the liquid is, it doesn?t change the level needed to be a quarter full, and 2) semi trucks don?t use gasoline anyway.
I think MIT might want to offer that guy a refund on his tuition if he agrees not to tell anyone that he went to school there.
A horizontal, cylindrical tank of any size, is 25% full when the fuel depth is 30% of the diameter and 75% full when the depth is 70%.
I’m a fuel truck driver. Having a chart that lays this out makes this stuff pretty easy.
I got the same thing, I think. This was a fiendish problem, but required no calculus for a precise answer…
This from an 8th grader at Duke School, in Durham, NC.
Nah… 13 inches would only get you about 69% full, not 3/4.
You cant calculate it that way as your miles per gallon will vary on how you are driving, speed/traffic conditions etc as well as on the terrain.
Your fuel mileage will vary depending on your load/traffic conditions and terrain.
You can not get an accurate measurement of fuel based on miles driven in a Big Truck. Mileage is affected by load weight, traffic, temperature, and elevation. The same truck will get better mileage in Ohio than it will in West Virginia. That is based on my having Driven a Tractor Trailer for a while. I watched my two math whiz kids fail to work it out with advanced math. I used a piece of graph paper, compass, and pencil to figure out a rough estimate. I got just under 6". One break down would cost more than fixing the gauge.
Here’s a little story for all you crazy kids.
A truck driver on a country road stops in front of a low bridge that his trailer is one inch too high to pass under. The police arrive with town planners and engineers in tow. A ten year old boy watching all the commotion of these ‘trained experts’ troubleshooting the problem said he had an idea but was dismissed by all the said experts. A cop suggested they back up all the traffic a mile, reverse the truck that same mile of winding road and detour down a side street. “Too dangerous and time consuming” someone said, the town planners suggest they get a smaller truck from a local company to hook to the trailer, thus lowering it. “That would lower the front of the trailer but not the rear” said someone in the group and the engineers suggest they jack up the bridge the one extra inch needed. To this the ten year old child chimed up and said, "who don’t you idiots just let some air out of the tires.
Always remember, work smart not hard and everybody knows something you don’t.
The issue with trying to solve it mathematically is that besides the personal satisfaction of solving the problem it really wont be applied practically. You’re going to round it off to the nearest 1/16" and at the end of the day, Rich is going to take the dowel and measuring tape in one hand and mark it with a permanent marker, possibly 2/16" wide if not more with the other, potentially placing the mark up to 1/4" off target.
Since Rich just wants to know where his 1/4 mark is so he knows when he’s down to his last 300 miles or so of fuel, then the easiest, most accurate and most practical solution is as follows.
The tank is probably a 100 gallon tank, so when the fuel level is at the 10" mark, that is 50 gallons sooooooo when fueling the tank, stop at the 10" mark and then add 25 gallons. That’s your three quarter mark. Providing you wait 30 seconds or so for the fuel to stop sloshing around in the tank before taking the measurement, the surface edge of the Diesel will be more accurate than any measuring tape. Of course you would have to either waterproof the dowel, or tape it before making your mark, just to account for any capillary action that might affect accuracy…
… and this solution would be applied whilst the truck was being refueled anyway, so both calculating the problem and applying it would have only added about 5 mins to Ricks schedule.
God, if I went to college, I’d actually be dangerous.
Happy problem solving folks.
Yours Sincerely
Mark Blake
Peter is my eighth grade math student at Duke School. He was looking for something interesting to work on. He’s 13. Pretty awewsome, I think!
Bill O’Connor, teacher
A lot of good thought processes; always liked people who listen to Car Talk. Three solutions come to mind immediately:
x^2+y^2=1
Solve in terms of y:
y = sqrt(1-x^2)
We don’t need to deal with half of the tank, we really only need 1/4 of it. If we put the circle on a cartesian coordinate system with the middle of the tank at the origin, we’ll only be looking at the 1st quadrant. Our integral will over y from 0 to z, where z is the distance from the middle of the tank to the top (or bottom). Now is where I really wish I had MathType.
In any case, you set that integral equal to 1/2 of the area of that quarter circle; in this case, pi/8. With some fancy footwork (change of variables from z to sin(theta) and then application of trig identities), followed by insertion of limits, you get:
sin(z)^-1+z*sqrt(1-z^2)/2 = pi/8.
With some easy application of mathematica/calculator, you end up with z = 4.03973. (the height from the midline of the tank)
If this is too much (trig is always horrible), why not just use a simplified version of calculus? All calculus does is take a summation of infinitely small rectangles under a curve; you can make those rectangles have finite values and come up with an approximation that will do in this case. Divide the x coordinates into 10 equal partitions (0,.1,.2,.3,…,1) and calculate the y coordinate at each location. Multiply each coordinate by .1 to get the area, and then determine where the summation of areas becomes greater than pi/8. The more subdivisions you use, the more accurate this becomes; then again, Rich is using a literal stick to measure things, so incredible precision is not a huge concern.
If you flat out hate math, do what I do for my car (whose gas gauge also has been 86’d). Look at a spec sheet for the mileage that the truck gets (or take a reasonable estimate), multiply by the capacity of the tanks, and you have your total possible mileage. Then subtract a “fudge factor” (city driving, inefficient engine, etc) and you have the limit of distance that you can drive safely. From that point, use the odometer and take fractions of that distance. Only restrictions are that you have to fill up every time you stop and reset the odometer, or take very, very careful logs of your distances driven and amounts filled up.
Trust me. I’m a rocket scientist. 
My fuel tanks are labeled: 101 gals cap., 93 gals usable. I figure I have at least 175 gals to use. I get between 5.5 and 6 MPG, so I figure 5 gals. 175 X 5 = 875 miles range. When I fill up, I reset the trip odometer to zero. When the trip odometer gets to about 750 miles, I know I need to start to look for fuel. Actually, I fuel most every day, because truck stops require a minimum 50 gal fuel purchase for a free shower. (Otherwise it is about $10.) To determine mileage: fill the tank, drive around for a while, fill the tank again. Divide miles driven by gallons used.
You can use the attached excel document to calculate the liquid in any given cylindrical horizontal tank. Stick the tank to get the level of the liquid. Enter the Length of the tank and the Diameter of the tank. Then go down the depth column to the height in inches of the liquid and read the volume in gallons. You can also make your own calibrated stick. For extremely large tanks you may need to extend the columns. The Auto Parts Stores sell a little reference book that has a simplified formula for calculating liquid volume of a horizontal cylindrical tank.
Hey, Kaliki.
You seem to know what you’re talking about, but you didn’t realize that
you answered a different question. The centroid of a semi-disc is
located slightly off the half-area-line that our trucker needs.
I mean half the area of the semicircle, which is a quarter of the full circle.
Anyway, it’s easy to get confused, and I think there was another post here
with the same mistake. Your number (4.2 inches rounded to two digits)
is only slightly different than the correct number of 4.0 inches, and
we may be tempted to ignore the difference. But it’s instructive to understand
the subtle difference between them.
In both cases we do some kind of balancing, but we do it differently.
The centroid is the center of mass, and we find it by trying to balance the weight;
I mean balancing in true sense of the word;
so, the distribution of mass/weight is very important;
specifically, the distance between each little piece and the centroid;
and if we do it right, we could balance a cardboard semi-disc on the tip
of a pencil placed exactly at the centroid.
But when we find the quarter-line, we simply balance two numbers:
the areas on either side of the line.
And it doesn’t matter what is the distance between our quarter-line
and each of the little pieces that we keep track of.
Actually, according to my Ph.D. in physics friend, the value is 35.04", or 7.008" for a 20 inch diameter tank.
Corrected: I meant %.
Actually, according to my Ph.D. in physics friend, the value is 35.04%, or 7.008" for a 20 inch diameter tank.
Hi, Bill.
You’re off (sorry). 5.96" would be a quarter tank. I’ve been punching this formula into programmable caluclators for years. (This isn’t the first time a similar question came into Cartalk.) Respectfully,
Jim “jameshorne” Horne
There is no need for integral calculus, just trig.
This formula tells how far up from the bottom of the wooden dowel you’ll need to measure for the quarter way point.
Formula: r-(r/2)sin(pi/4), where r = radius
Answer: Given r = 10in, then the quarter way point is 6.46446609in from the bottom of the dowel.
Trust me, I’m NOT a rocket scientist 
with about six feet left in a tank, that gives approximately a quarter tank [could be done with calculus, but it isn’t necessary)