Fuel Gage Problem

[Jim Brier]

An 18-wheeler has a broken fuel gage. The driver wishes to measure the level of fuel in his cylindrical tank by using a stick. The tank diameter is 20 inches and the fill hole is directly on top. He knows that 10 inches is half a tank. He would like to know how many inches on his stick is one-quarter of a tank. Obviously, it is not 5 inches because the tank is cylindrical.

[Bozeman]

Are you just asking us to do trigonometry?

[JTG51]

Yes, I think he is.

Jim, it's too bad that most of your posts come in the form of math problems these days. Your contributions to the mid/high stakes forum are missed.

By the way, I really liked your most recent card player article (on playing high limits). I doubt I'll ever play that high, but I found it very interesting.

[Ed Miller]

I'm assuming you mean that the tank is "laying on its side" rather than standing on its base (or else 5 inches would be the answer).

If Theta is angle between the edges of the tank at gasoline level (at quarter tankness) and the center of the cylinder:

1/4 pi * R^2 = Theta/2 * R^2 - R^2 sin(Theta/2)cos(Theta/2)

Theta/2 - sin(Theta/2)cos(Theta/2) = 1/4 pi

The number of inches X for a quarter tank is then

X = R (1 - cos(Theta/2)) where R is 10 inches

Blah.. I have a meeting... [img]/forums/images/icons/smile.gif[/img]

[Homer]

Does the trucker have a computer? I came up with two ways of doing the problem, but for both I ultimately needed a computer to reach the final solution.

Method 1:

I used some trig and came up with the following formula:

{5/9*pi*cos-1((10-h)/10)} - {2*(10-h)*sqrt(20*h+h^2)} = 25*pi

I used the Goal Seek tool in Excel (basically plug this formula into cell A2 and h into cell A1, then tell it to set A2 equal to 25*pi by manipulating the value of h in cell A1) to find a solution to this equation.

With this method I got an answer of 8.04.

Method 2:

A horizontal "slice" of a vertical (circular) cross-section of the cylinder has an length of sqrt(20*h + h^2). If these "slices" are integrated from zero to x (with x being the height filled with gas) and this is set equal to 25*pi, solving for x will provide the solution.

I plugged this function into "The Integrator" and got some crazy ass function spit out at me. I guess I could plug this function into Excel and solve the same way as I did in my first method, but I don't feel like it.

...in any event, there is probably a ridiculously easy way to get the answer to this problem that I didn't think of. You probably didn't want us to use a computer anyway. [img]/forums/images/icons/smirk.gif[/img]

-- Homer

[Mano]

Assuming the trucker knows how much his tank holds, fill the tank to halfway (which can be done using his stick), and then add another quarter tank, thus making the tank 3/4 full. Now measure the height on your stick, and the height for the 1/4 full tank will be 20 - the height for the 3/4 full tank.

[JTG51]

There's a problem with that solution.

After filling the tank half way, you want him to add another 1/4 of a tank. If he knew how to add 1/4 of a tank, he wouldn't need to go through the process you described. He could just add 1/4 to begin with and measure that.

[Mano]

The problem with that would be, either he would have to run the tank completely dry (which would be bad, and difficult to time), or there would be some undetermined amount of gas left in the tank.

[JTG51]

That is a good point. But, since we only seem to be interested in measuring quarters of a tank, it doesn't need to be extremely accurate.

Of course, I don't think any of this matters. I don't think Jim is looking for a common sense solution, he's looking for the math. At least I think so.

I still like our ideas better. [img]/forums/images/icons/smile.gif[/img]

[Munga30]

You don't have to run it dry.

Fill the tank to 10 inches, which you know is half a tank.

THEN add the 1/4 tank, measured at the pump in gallons.

THEN measure with the stick and subtract 10 inches.