Hi Guys,
As it turns out, I had to do this calculation some time ago to figure out how much oil I had in a cylindrical underground storage tank. My solution is attached.
Al2525
This solution is updated November 17, 2010 to show the correct equation. If you downloaded the prior solution, delete it and use this one. Al2525
Going back to the last 3 steps of the solution… how did you get ?? What are the steps in inputting it in a calculator to get a value of 1.155? I didn’t learn to do that in my trigonometry class.
What kind of 18 wheeler uses gas?
Dear Trucker Rich:
Here is a simple solution.
Your tank is 20 inches in diameter (radius equals 10 inches). Measure the length of your tank and calculate its volume. Let’s say your tank is 30 inches long.
Calculate the volume in cubic inches (Pi times R squared times 30). The volume is 9424.77 cubic inches.
Convert volume to gallons with one cubic inch equal to .00432900433 gallons. That gives a tank capacity of 40.8 gallons.
One tenth of a tank is then 4.08 gallons.
Using your 20 inch measuring rod mark off the mid-point at 10 inches. Fill the tank half-way using the rod to measure.
Re-set the gas pump to zero and pump in 4.08 gallons. Put the measuring rod in to see where the fuel level is and mark it as 0.60 (6 tens of a gallon). Go the same distance down the rod and mark 0.40 gallons.
Re-set the pump to zero and add another 4.08 gallons. Mark the 0.70 and 0.30 gallon positions on the rod.
Repeat the process until the tank is full.
The result is a measuring rod marked off in one-tenth of a tank increments. Better than just knowing the one-quarter, half, and three quarter volumes.
Hello I am a Civil Engineer technician and I am providing a graphic solution to your dipstick question, see attachment. P.S. Great show.
That’s a nifty little graph, but I think we’re all more interested in the CORRECT solution to the problem. Not an answer that’s off by more than an inch.
A lot of people think they have given an exact answer. Since all the heavy math answers have neglected the rounded ends of the tank, I am going to give an answer that neglects the rounded SIDES! This thought experiment will give the answer in 60 seconds, and just uses rectangles and circles.
Imagine a tank filled to the 1/4 volume. Now freeze the fluid in place, and rotate the tank 180 degrees and again fill it to the 1/4 volume and freeze the fluid. You now have a tank 1/4 filled at the bottom, and 1/4 filled at the top, leaving an air space in the other half.(Draw yourself a picture.) The air space is now 1/2 the volume of the tank. The air space is “essentially” rectangular, with sides of 2R and 2(R-h) where h is the desired dipstick mark. That sentence in an equation is: 2R2(R-h) = (1/2) piR*R (7th graders can solve this) and you get h= (R - pi/8) = 6.07". It is a slight overestimate, but it gives me just a little leeway to get to the next fuel station.
I will follow my 7th grader solution by a 4th grader solution.
Although this discussion is slowing down, it’s good to see new posts.
Both solutions posted by oldnotdead are very good. Bravo!
But, and not to take anything away, they aren’t new.
After all, there are more than 200 posts now,
and most people cannot take the time to keep up.
The first solution (a very clever approximation) has already been posted
separately by two persons. You get extra credit for pointing out that the
slight error is on the safe side.
The second solution has also been posted (once) before.
This time you get credit for being concise.
The first post was quite long and attracted harsh (but unnecessary)
criticism from Josh (jt1979).
Although this solution is ingenious, it only seems practical in principle.
In practice, it requires accurate means of weighing.
The pizza cardboard doesn’t have to be the same size as the 20 inch cross section
of the tank. Any smaller circular size is just as good, although the smaller
it is the more precise the weighing scale needs to be. So, large is good.
Congrats to all the math people. My solution is more practical, because, really, do the math on what percent of the population can do the math I’ve seen in the “real” answer . . .
The guy knows his truck. Drive it until you’re sure it’s under 1/4 full. Mark the dowel. Fill up the tank. Subtract the added fuel from the tank’s capacity. You now know what the dowel marker indicates, let’s say 1/8.
Drive the truck until you’re back down to the dowel mark. At the gas station, put in 1/8 of the tank’s capacity (i.e. 1/8 of the truck’s listed fuel tank capacity). Mark the dowel–that’s a quarter tank. Fill another quarter tank of fuel. Mark the dowel–that’s half. Fill another 1/4 tank’s worth–that’s 3/4 of a tank.
Of course, this answer is not accurate to the drop, but is that what the trucker needs?
The remaining volume equals H/D x Total tank volume, where H=height on dipstick, D= tank diameter.
Answers are shown on page 384 of Tom Glover’s fine little masterpiece that everyone should own; “Pocket Ref” from Real Goods 1-800-762-7325, published by Sequoia Publishing of Morrison, CO.
The table gives many more values, but here are some low Ratios vs Depth Factor H/D values.
0.04=0.01348, 0.06=0.02451, 0.10=0.05202, .0.20=0.14235 0.30=0.25230, 0.40=0.37354
3rd grader solution:
OK, OK, I didn’t read all the posts, and my 7th grade and 4th grade solutions had been basically presented in slight variation. So now, I’ll try a 3rd grade solution. Maybe it also has been presented.
Ha ha ha!! Very nice!
Even 3rd graders can integrate, because it comes down to counting little bits.
If they do the counting for a square and for a circle,
they will get a good estimate of pi.
They’ll have to wait a few more years to know what it all means, etc.
But isn’t nice to see how big-and-nasty problems can be reduced to basic things.
And, after all, in real life (of hard science, etc) we deal mostly with nasty
integrals which can only be done numerically. So, all fancy math eventually
comes back to basic arithmetic; it just that we do it with very fast computers.
Let this website do it for you! http://www.handymath.com/cgi-bin/arc18.cgi
I thought he said his tank is 22" in diameter. That yields a little over 6.55" fo a quarter-tank.
Hey Guys ! I don’t know much about big trucks or math either but here is a solution to your puzzler that doesn’t require diesel on the hands OR invoking calculus. If the tank diameter is 20 inches a quarter tank will be 5.96 inches deep, please check the attached file.
IOn
Thanks for the comment and suggestions. I’ll pass along your challenge to Peter. I wonder what he will do with it?
Bill
Ion –
As it turns out, I was just recently working on this one. Strangely, what sparked my interest is that my second grade teacher told me one day that splitting a circle in half parallel to the base was impossible; I’ve worked on the problem on and off since.
Just recently, I observed that the number of trianglular units filling a triangle was proportional to the square of the height of the triangle. Your rhombus tank appears to be just this, as the rhombus is a triangle on either side of the 1/2 mark. For any triangle you want to fill up half-way, the fraction of the height from the top is ~.707, or the square root of one-half. From the center, it must be ~.292
I’ll get the next ones ASAP.
The Cartalk home page shows some solutions to this problem, including one that is wrong. This is the one that uses the formula for the centroid of a semicircle. It took me a while to understand why this correct formula does not give the correct answer! I’ll leave this as a question for others to ponder…
Draw circle 1 with diameter.
Draw circle 2 with same diameter as circle 1.
Move circle 2 up and down untill area A is equal to area C.
Draw chord shown
At this point area under chord is 25% of total area.
Measure X ( about 1.25)
Measure Y (About 4 )
Height on stick = 1.24 / 4 or 31% of diameter of tank.
Trucker Rich can get a pretty good estimate all by himself at the fuel pump:
No muss, no fuss. And it will be plenty accurate for a trucker who uses a stick in place of a fuel gauge.