The area of the full tank cross-section is pi * 10^2 = 100 * pi
Draw the circular cross-section centered at the origin.
The formula for a circle is x^2 + y^2 = radius^2; so y = sqrt(radius^2 - x^2)
You want half a circle’s area to equal 1/4 * 100 pi = 25 pi.
The integral from x=-5sqrt(2) to 5sqrt(2) of sqrt((5*sqrt(2))^2 - x^2) dx = 25 pi
Therefore the depth measured on the stick should be 5 * sqrt(2) = 7.01 inches for 1/4 tank.
You guys are making this way more difficult than it needs to be. It can be solved without math by using a piece of cardboard, a piece of string, a pencil, scissors, a utility knife, a couple of plastic bags and a yardstick. (If you have a scale capable of measuring grams, it will be useful, but it?s not necessary.)
On a piece of heavy cardboard draw a circle of diameter d, d being the internal diameter of the tank. You can use the string and pencil to scribe this if you don?t have a compass hanging around.
With a straightedge draw a line through the center of the circle. Draw another line perpendicular to that one through the center as well. You have now divided the circle into four equal quadrants.
Cut the circle out of the cardboard, then cut the circle in half along one of the diameter lines. You now have two half-circles. Cut ONE of the half circles in half along the radius line, so you now have two quarter circles and one half circle. Throw one of the quarter circles away.
If you don?t have a scale, make one by taping identical plastic grocery bags or ziplock bags to each end of the yardstick. Suspend it with a string tied to the 18 inch mark; it should balance pretty well. Put the remaining quarter circle of cardboard in one of the bags.
Return to the half-circle; start cutting strips of cardboard off the half circle, starting on the round surface opposite the straight edge. Cut off strips parallel to the straight edge, perpendicular to the radius. Put the strips in the empty plastic bag. As the weight in the bag with the strips begins to approach the weight of the bag with the quarter circle, make the strips thinner. Once the scale is level, the amount of cardboard in each bag is the same, and you can measure the distance from the diameter edge to the newly cut edge. Or, you don?t even need to measure it; Rich can lay his dowel with the half full notch right on the diameter edge, and make another notch at the level of the new edge. That interval represents one quarter of a tank of gas, so if the half-full mark is accurate, so is the quarter-full mark.
If you have an accurate scale, you can of course weigh the quarter circle. Then cut off enough strips from the half circle to equal that amount. If you have access to an accurate lab balance, this entire exercise could be done with paper. Cardboard, though, would probably provide enough mass to make the experiment feasible with a low resolution scale such as a postage scale.
I took quantitative analysis 4 decades ago. The only thing I remember about it is a trick related to this: to calculate the area under a blip on a graph (such as you would get with gas-liquid chromatography) just cut out the blip and weigh the paper on a Mettler balance. Then weigh a known area of paper, and you can calculate the unknown area by the old means & extremes. (This is one problem where a slide rule might give you the answer faster than a calculator.)
No need to measure any distances. No need for calc.
Given: Dipstick marked correctly at 1/2 tank
Steps:
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.To satisfy my curiosity in this matter, I called the local Kenworth dealer's service department to ask about the unusable fuel. Now, I asked about our twin 2007 KW T800?s, with dual 100 gallon tanks, and he told me there?s no more than a half an inch between the bottom of the tank and the fuel line. In fact, he told me it?s not even worth draining because that fuel is getting mixed up all the time. I didn?t ask about older trucks, but my younger brother, who drives those trucks and has driven other trucks for other farmers (including some pretty old ones), said that he thought old trucks had fuel lines coming out of the very bottom of the tanks.
So, what?s a half an inch in the bottom of the tank? Less than 0.7% by my calculation. I?d say that?s negligible enough that?s it?s not worth worrying about for the purposes of estimating fuel.
I suppose now you?re doing to complain about the inaccuracy caused by the fuel that?s displaced by the dowel in the tank.
I didn’t read through every word of that nonsense, but from what I did, it certainly doesn’t sound less difficult than using a little simple geometry and trigonometry to calculate the area of a pie shaped partial circle, a triangle, and the total circle area. In fact, it looks quite a bit more complicated and time consuming, and less accurate.
ps Also no need for geometry, trig (or even algebra for that matter); or cardboard, strings, pencil, compass, pringles cans (loved that one BTW), factor in “useable gas” in the lines, run the tank dry, etc.
C’mon guys, this man is a trucker. Just keep it simple and git’er’done…ain’t got no time to be running out of gas, getting towed, precisely measuring distances, cuttin cardboard, contacting the math department at State U, eating a whole can of Pringles while driving,etc,etc,etc
pps My qualifications: in my younger poorer days, I ran out of gas more times than I care to remember. Sucked every time. I got A’s in algebra, geometry, trig and B’s in Engineering (not “sandbox” ie “Business”) Calc classes. Sucked every time as well. All these math-head type answers have resurrected my recurring nightmares of showing up to math tests naked and unprepared. Thanks a bunch. Good thing I could solve this with virtually no math as I’m so old now I can barely remember how to multiply and divide.
Do this at a gas station. Since you know the capacity of the tank, drain it completely. Use the flow totalizer on the gas pump to fill the tank to 25% of the known capacity. Next, stick the dipstick in the tank to mark the level on the dipstick. Then repeat for 75% of the known capacity.
I also agree that this can be solved in 2 dimensions using the cross section of a cylinder. And due to symmetry, to simplify, you can divide this circle into quarters. To get the area at the bottom of this circle with the flat top, (the level of the fuel) from the measured distance from the bottom (M) ? the yard stick,
First: using M, calculate the angle that the fuel level and circle center will make,
then find the area of this pie shaped piece,
then subtract the area of the triangular part of the pie shaped piece (the top part), and what?s left is the area on the bottom, the fuel.
Then divide this area (of the fuel) by the area of ? of the circle, the quotient is the fraction of the tank that the fuel occupies.
If you keep the angle in radians, then no conversion from degrees is necessary.
The answer agrees with the ? tank level of a 20? tank to be 5.96? or 0.298 dia.
I also made a table from 0 to full in 1/8 increments. What?s interesting is the ? full yardstick measurement for the cylindrical tank is LOWER than if it was a rectangular (linear volume with height) tank. One way to look at this is that if the cylindrical tank was at 19? (one inch from full) hardly any fuel has been used since there?s very little area up there. Conversely if the measured fuel level is 1?, there?s hardly any fuel left.
I?d like to see that derivation using integration by parts by needsglasses
I suppose now you?re doing to complain about the inaccuracy caused by the fuel that?s displaced by the dowel in the tank.
No, I am not. Why would you assume I would do that, when all I am doing is asking questions and trying to be as precise as possible?
It comes as no surprise to me that a 2007 truck would be that way. After all, low sulfur diesel fuel and ultra low sulfur diesel fuel have been around long enough that you have simply proved my point that we need to know the year, make, and model of the truck in question. Trucks like these can be on the roads for decades. In fact, for a 2007 truck, the law would require that only ULSD be used in it. Now if you said that about a 1997 truck, that would comes as a big surprise to me.
Why am I the only person asking for year, make, and model? Is this not an automotive forum?
The purpose of the exercise was to show a non-mathematical solution. The premise is valid: that on a uniform material like paper, you can use the weight of a known area (in this case the quarter circle) can be compared with the weight of an unknown area (the strips of cardboard being removed from the half circle). When the weights are the same, the areas are the same as well. Rube Goldbergish it may be, but nonsense it isn’t.
I think your answer is interesting, arleyg, because you are using calculus! Slicing the circle up into thin strips and weighing them is effectively the same thing as taking an integral, which adds up tiny slices of the area under a curve. I bet there are calc books out there that have a picture much like what you describe in the chapter where they introduce the definite integral.
I believe Rich only said that his dipstick was vertical and didn’t say exactly where the filler hole was. If it is not at the very top, the dipstick either won’t touch the very bottom or it will be at an angle to the vertical. Even if the filler hole is at the very top of the tank (which I question having looked at dozens of tanks at trucker.com), that doesn’t mean you can fill the tank to the top. Maybe Tom and Ray can call Rich to resolve this paticular issue.
While I’m enjoying the debate, don’t you think you’re all getting a little carried away here. Yes accuracy is important but all the truck driver wants is to have a mark that comes close to the 1/4 full level. You could debate the usable/unusable fuel; whether the tank actually ever gets COMPLETELY filled; what is the diameter of the dowel he’s using - does it have a flat end or curved? All of these and more could affect the EXACT answer. What the trucker really wants to know is that at 6" he has very close to a quarter tank of fuel left.
For what it’s worth (not much): one iteration of Newton’s method on
f(x) = arcsin(x) = x*sqrt(1 - x^2) - pi/4
with x=.25 as an initial guess gives a=.4, and successive iterations don’t improve on that much. That gives the same answer as many other here: .4 * 10" = 4", so he should mark the dipstick at 6" for 1/4 of a tank and 14" for 3/4.
Callibrating a dip stick for an 18-wheeler fuel tank
I had to laugh today as I heard you struggle
to figure out how to mark a dip stick for the
caller who needed to measure diesel levels in
his 18-wheeler. Apparently I’m not the only
one who ran away from “The Calculus” after
college!
Any general aviation pilot knows how to solve
this problem. And we don’t need no stinkin’
Calculus!
The fuel gauges in my 1971 Cessna Cardinal
are, like in most small planes, completely
unreliable except when they read “full” or
"empty."
It’s really bad when they read “empty” so we
are trained NEVER to rely on the fuel gauges
and are required to visually inspect the
tanks before takeoff. But the tanks are
irregularly shaped, so just eyeballing the
fuel level isn’t good enough. (About one GA
pilot per week gets this wrong and finds
himself suddenly flying a glider.)
I use a homemade clear plastic dip stick with
which I can determine within about one-half
gallon how much fuel remains in each of my
two 25-gallon tanks. Here’s how you make one:
Select a dip stick that is long enough to
measure “full” in the tank. Mark the
dipstick in even increments along its length
and number the marks. The actual dimension
of the increments doesn’t matter – just pick
a reasonable size. (The length of the stick
for my 25 gallon tanks is about 9 inches, and
it’s divided into 1/2 inch increments.)
On a sheet of ordinary graph paper, mark the
vertical axis in gallons (I mark increments
of 1 gallon) up to the capacity of the tank,
and the horizontal axis in as many increments
as you marked on the dip stick, plus a few
more.
Drain the tanks as near empty as possible. Then:
Fill a 5-gallon gas can with gas – exactly
5 gallons – and empty the can into the fuel
tank.
Stick the dip stick in the tank, and note
the numbered mark on the stick.
Plot the dip stick mark on the horizontal axis.
– If you were able to empty the can into
the tank, plot on the vertical axis the
number of gallons added since the beginning,
refill the 5-gallon can and repeat
– But if you could pour only a fraction
from the can into the tank because the tank
is now full, locate your plot on the vertical
axis against the known fuel tank capacity (in
my case 25 gallons), and stop.
Now create a curve from the final plotted
point that is parallel to the curve created
by the prior points. This curve will give
you the actual fuel levels the correspond to
the marks on the dipstick.
The trucker wanted a stick that he could use to measure, in particular, the last 1/4 tank level. Here’s how:
Hanford Reed
Honor Graduate…school of common sense!!
Or, I guess he could buy a new truck!!
Wow, imagine that. I never took calculus. (English/premed major.) Maybe for just one moment I was channeling Leibniz and Newton…
Big Bruce and SKRA…etc? One word: wrong.
Have you guys read the other responses - including mine - before submitting? Just curious.
I verified this answer using high school geometry/trig (cause really it takes too much brain power to remember calc)
1/8 tank is a bout 3.6"
I think calling it nonsense was a fair statement based on your first sentence - “You guys are making this way more difficult than it needs to be.”
Not only is your method much more time consuming and difficult than a little simple math, but it is certainly prone to errors in making those measurements. Make a scale by taping grocery bags to the ends of a yardstick?
The principle of what you’re describing may be sound, and it’s certainly a unique and creative take on the problem, but I don’t think it’s at all easier, and certainly not as accurate as calculating it mathematically. I didn’t use calculus to do it - my degree is in accounting, not math, and while I think I understand the math puzzles better than the average person with an accounting degree, I don’t remember much, nor did I like, calculus. I just used a little geometry and trig, and the “goal seek” function on every accountants favorite piece of software, Microsoft Excel.