This week on Car Talk, we heard from Rich, who was piloting his 18wheeler through Carthage, Missouri. His problem? He's truckin' without a fuel gauge  and he sure doesn't want to be stalled on the side of the road with two tons of frozen turkey wings thawing before their time.
What's he got for a tank? Unfortunately, it's not a simple cube. For that matter, it's not even one tank  he's got two "saddle" tanks, one on each side of the cab. They're both cylinders, strapped lengthwise under his cab. Right now, he uses a wooden dowel to get an approximate sense of how much fuel he's got left.
You can hear Rich's call to Car Talk right here.
So, here's his question: how can he figure out when he's only got a quarter of a tank of fuel left? And while we're at it, how about threequarters of a tank?
Think you can figure it out? (Hint: anyone remember integral calculus?)
Let us know how you'd solve this problem!
Yours in getting frozen turkey wings to the market on time,
Tom and Ray Magliozzi
Click and Clack, the Tappet Brothers
No math practical method:
1 Fill tank (20 inches)
2 Drive to half tank ( 10 inch mark), Miles = x
3 Refill tank
4 Drive Half of x distance
5 Mark dipstick fuel level for 3/4 tank
6 Mark dipstick for 1/4 tank, this distance is the same amount below half as the 3/4 mark is above half
You are over complicating the math. Use the trip odometer. Set at zero. Drive until the dipstick is at the half tank mark. Divide the milage by two. Add to the trip odometer and drive until this milage is reached. This is the 1/4 tank level.
The 3/4 level is the same distance from the half tank level.
http://mathworld.wolfram.com/CircularSegment.html
this shows the incedibly complicated formula to calculate area of a sector of a circle…
the final answer is 5.96" from the bottom of the tank is 25% of the Area.
The fuel tanks are joined, on most models of tractors, by a fuel line between them, so that when the fuel gauge, if it worked, reads the fuel level in BOTH tanks. Assuming each tank has a capacity of 250 gallons, when the dipstick or the gauge reads a half a tank, then you have a total of 250 gallons of fuel to drive on, the same as one complete tank. That being said, a quarter of a tank would still leave 125 gallons of fuel and an eighth of a tank would leave 62.5 gallons of fuel. When the truck is fueled, as close to empty as is comfortable, only put in 125 gallons, wait a few minutes for the fuel levels to equalize in both tanks, and take a reading on the dipstick. If this were me, I would just take it in somewhere reliable and have them check out the fuel sensing system. The problem could be in the gauge itself, the wiring, or the fuel sender in the tank (the most likely candidate). Being a professional driver I would not take a chance in misreading a dipstick. All this is also assuming this is an older tractor. Newer model tractors have a fuel sensor in each tank and the ECM reads the resistance being sent to it, eliminating the need for a crossover tube between the tanks, each tank having it’s own fuel pump.
Geeze guys! Are you really that dumb? Run one tank dry, switch to the other, drive to a gas station, fill the dry tank with 1/4 of the stated volume. Stick the stick in, make a line where it’s wet. Continue to fill for 1/2 and 3/4 if desired. You don’t need no stinking calculus. My respect for your common sense is at the warning level.
A couple of years ago, I googled “Liquid Volume in a horizontal Cylindrical Tank” or something similar and got 2 or 3 different formulas. The easiest one for me to reduce to a spread sheet from ASK DR MATH, was:
Consider a circle of radius a (the end of the tank)and imagine that the liquid has reached height h, measured from the lowest point on the circle. Note that 0<=h<=2a. The area A of the segment of the circle covered by the liquid is:
A = pia^2/2  a^2arcsin(1h/a)  (ah)*sqrt(h(2ah))
The volume of liquid is just A*L, where Lis the length of the tank.
To check the calculation, a verticle tank volume of the same dimensions is much easier, and the volume matches within a gallon out of 240 gallons, which is close enough for automotive work.
I also run a Big Rig once in a while and am concerned about legal weight limits. I have a spread sheet that considers 2 identical tanks on the truck and equates liquid level, gallons and weight. If I can get Richard’s tank dimensions and number of tanks, I will email him a printout of what he has.
Hi Guys! Listening to your show makes me laugh a lot, especially valuable in these times. My user name reflects owning a Prius, which has no gears that shift. I’m actually a major gear head. How did you both graduate from MIT and have a sense of humor left?
As one who used to have oil heat, I am familiar with turning dip stick readings in a horizontal cylindrical tank into percent of capacity. If you visit:
http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002.Fall/Wright/6690/essay%201/essay1.html
you will see the full solution, no integral calculus needed! The dip stick to gallons table at this site can be adjusted by simple ratio to Rich’s tank capacity and he will have is answer to 13 decimal places! As you correctly asserted, this is a two dimension problem.
Once again, I really enjoy your show.
Start with unit circle equation: x^2 + y^2 = r^2 = 1.
f(x) = y = (1x^2)^1/2
To find the point where x equally divides the area under the curve from x=0 to x=1,
integrate f(x). Evaluate it from x=0 to x=a and set it equal to the same equation evaluated from x=a to x=1.
You end up with 2/3 (1a^2)^3/2  2/3 = 0  2/3 (1a^2)^3/2
Then solve for a.
a=0.392 from the center, or midline of a tank with r=1".
Since this is proportional for a tank with radius of 10" or a diameter of 20", I know that the 3/4 and 1/4 tank mark is located at 10" +/ 3.92"
Therefore,
Full tank = 20"
3/4 tank = 13.92" or for all practical purposes 14"
1/2 tank = 10"
1/4 tank = 6.08" or for all practical purposes 6"
You are over complicating the math. Use the trip odometer. Set at zero. Drive until the dipstick is at the half tank mark.
I think you’re oversimplifying things  like the spelling of the word “mileage.” What do you suggest he do, pull over every hour to measure it? That’ll be great for mileage! His tank is 20 inches in diameter; I wouldn?t think it?s any longer than 60 inches (and that?s probably an overestimate). Two tanks that size would give him about 160 gallons total. At 8 miles per gallon and 65 miles per hour, he?s burning 5% of his fuel every hour.
Geeze guys! Are you really that dumb? Run one tank dry, switch to the other? My respect for your common sense is at the warning level.
He?s driving a truck, not flying an airplane ? I don?t think you can select fuel from only one tank. In fact, I read something that suggested if you park a truck with dual fuel tanks on a hill, where one tank is lower than the other, fuel will crossflow from the higher tank into the lower one, and can cause problems.
I think the reason this problem is interesting to many on here is BECAUSE of the mathematical component, and challenge therein. I?m guessing you?re not one of us whose ears perk up when they announce that the new puzzler is mathematical in nature.
There is several post out on this, so this is a copy of my reply from a earlier post
The dumb down way to figure out where to mark the dip stick for the truckers gas tank is.

Drain the tank empty.

Find what the published volume of the tank is.

Divide the volume the tank can hold into quarters.

Add one quarter of the volume of the fuel to the tank at a time, marking the stick after each quarter.
?Assuming each tank has a capacity of 250 gallons, when the dipstick or the gauge reads a half a tank, then you have a total of 250 gallons of fuel to drive on, the same as one complete tank?His tanks are only 20 inches in diameter ? they would have to be more than 15 feet long to hold 250 gallons each. Out of curiosity, I found a company online where you could order a custom truck fuel tank. They didn?t offer anything larger than 200 gallons, and according to the Peterbilt website, a new 389 can be had with tanks up to 150 gallons. I don't think you're getting 250 gallons tanks on a truck, and certainly not with a 20 inch diameter.
Well, consulting the table of integrals, I found that you would have to solve H(100H^2)1/2 +arcsin(H/10)=157.0796. Not remembering how to solve for H with arcsin in there (Its has been over 30 years since I took calculus), I solved for H iteratively. First Guess was 3, which was too low, then 4, then 4.05, and finally, I got 4.04, and figured that was close enough. The notches on the stick should be:
5.95" for 1/4 of a tank, 10" for 1/2 a tank, 14.04" for 3/4 of a tank, and full naturally is 20."
OK, if you do the math, the answer is he needs to put his mark for 1/4 tank of gas about 6 inches up the stick. (The 3/4 mark will be 14 inches up.)
But to hell with math! There are many more creative ways to get the correct answer. Your own lamebrain attempt (go around at a truck stop till you find someone with exactly a quarter of a tank and measure theirs) is likely to get the poor guy beaten up. “Hey, mind if I shove my stick in your tank?” is not a question I would wander around a truck stop asking. I’m just saying. At best your solution would be hit or miss and take potentially a long time.
But consider this:
Drive on one tank until it becomes empty. You know how much it holds, so just fill the tank that number divided by 4, then measure. So if it holds 100 gallons, put 25 gallons in and then use your stick and make your mark. Easypeasy, no?
And by the way … I seem to remember you guys positing this exact same question around 20 years ago. Yeah, really! Your material is so old you’re forgetting it yourselves. Oh wait, that means I’m … umm … getting old too … uh, never mind.
I like the calculus method better because:
(1) it took less than 10 minutes to figure it out,
(2) I could do it in the comfort of my warm home,
(3) I didn’t have to drain a fuel tank which is not as simple as it might seem,
(4) I avoided unnecessary handling of diesel fuel  by the way trucks don’t use “gas,”
(5) it brought back fond memories of my high school calculus teacher from 23 years ago.
Work smart  not harder.
I’m not as smart as everyone else . .and forgot all my math . . if you measure down from half a tank, the area of a rectangle is pretty much the same as the area of the piece of a circle that is close to the diameter. If the area you want is 1/4 of the area of the tank circle, then a rectangle of 20 inches (diameter of Rick’s tank) times ? roughly equals 1/4 of the area of a circle. So . . .if pi times R squared is the area of the circle, then pi (3.1416) times 100 (radius of the 20" diameter of ricks tank squared) is 314 sq inches. Divide that by 1/4=78.5 sq inches . . . a rectangle with one side of 20" (the diameter of rick’s tank) and the other side of 3.925 is roughly 78.5 . . . .
If Rick measures down a bit less than half way plus 3.925" or 13.925" (call it 14 inches, I can’t see that well) inches from the top, he’s got roughly 1/4 of a tank left . . close enough to try and read on the side of a wood dipstick on the side of the road in a darkening evening sky.
Problems with using a dipstick in general:
 Even a 20 year old can’t see THAT well on the side of a stick.
 The truck might not be level front to back.
 The fuel wicks up the side of the stick, giving you the feeling there is more in there than the stick actually measures
I ran a '54 cornbinder (Int’l Harvester) in the woods once many many years ago, and we were too tired n too cold for math, so we figured sumthing less than 3/4 down was close enough to running out before we got back to the road, let alone a filling station . . at least that way we could wait until someone ELSE came by to give us a lift to fuel, and not foul up the injectors.
The stick should be calibrated as follows:
~ 6" from bottom = 1/4 full
10" = 1/2 full
~ 14" = 3/4 full
20" = full
It does involve calculus, but also reduces to a horrible equation for overeducated #@&%heads with too much time on their hands. Imagine a pair of 10" adjacent pizza slices that are each ~ 62.2 degrees (~124.4 total degrees when together) that are touching the bottom of the tank, with the adjacent line between the slices being vertical. The center of the chord formed by the edges of both slices will be ~ 6" from the bottom. The area of one pizza slice  the area of the triangle above the chord of one slice = the area of the chord of one slice (between horizontal chord and bottom of circle), and this = 1/8 the total crossectional area (or 1/4 the area for both slices). If THETA is the angle of one pizza slice, the equation reduces to:
THETA((180/pi)*cos(THETA)*sin(THETA))= 45,
and you have to solve for THETA, which can be done by trial and error for expediency. Now I’ve completely forgotten what the puzzler was. On the other hand, this guy could install a camera in his gas tank and recall it on his smart phone using an app. Then when he gets stopped by the cops for appearing to be texting, he can say, “It’s right for you to be concerned, officer, but I was only looking at my gas gauge–no, really.”

Get a Pringles can and eat all the chips

Fill the can 1/4 up with water

Put the cap back on

Tip the can on its side
The Pringles caps are usually translucent enough so you can probably see the water level inside. The ratio between the height of the water and the height of the inside of the can will give the ratio for the dipstick height (relative to a full tank).
This is a lot easier than it might seem.
First, fill the tanks up.
Note the mileage on the odometer.
Next, drive until the you think that the tank is just almost half full.
Make a half full mark on the rod at the half the diameter of the tank.
Measure with the rod to make sure you are close.
If not, drive a little further and check again.
Repeat until very close to the half full mark.
When you are as close as you can get, note the mileage on the odometer.
Suppose you have driven 1000 miles on one half tank.
Just divide that number in half and drive 500 miles further.
Now take the rod and measure the tank again.
Make a notch here, because this when the tank when it is 1/4 full.
To calculate where the 3/4 full mark is,
Add the distance of (the half mark to the 1/4 mark) to the half way mark on the top part of the rod.
It’s a piece of cake, not pi.
Stewart Brann