A Truck, a Box, and More Than 8th grade Mathematics!

Jan 17, 2011

RAY: This is from my lower mathematics series. This was sent in by an anonymous physicist who clearly doesn't want any of his colleagues to know that he listens to Car Talk. He writes:

'Sometime ago you entertained a caller who wanted to know how to measure the fuel level in the cylindrical tank of his diesel truck. These tanks are cylinders that lie on their side and the filler is on the top. Specifically, he wanted to know if he inserted a broomstick through the tank's filler opening, where on the stick should he put the ¼ full mark?

'His tank was 20-inches in diameter. While he knew that the half-full mark was at 10 inches, he was doubtful that the 1/4 full mark was at 5 inches. You two bozos agreed with him and after fumbling around for a few minutes, determined it would take more than the 8th grade mathematics at your disposal to figure out the answer.

'Well, a few weeks ago, my wife and I were traveling from DC to Wilmington, North Carolina, by car. While stopped at a roadside diner, I told her the story of your mathematical misadventure. I noticed a fellow a few stools down listening intently. After a while he interrupted us and said, 'I happen to be a diesel truck drive myself and while I don't know the answer, I think I might have something in my truck that can help figure out the answer. '

"'You do?'

''Yeah. It's something that came with last night's dinner,' he said. With that, he sprang from his seat and two minutes later, returned with a rather large box in his hands. He then produced a jackknife from his belt, asked our waitress for a pencil and with a surgeon's precision, he got to work."

The question is what did he do to determine where the mark should be on the stick to signify the ¼ full mark on the tank?
Answer: 

RAY: Here's the answer. The other fellow at the diner was a truck driver, and what did he retrieve from his truck? A pizza box! Inside the pizza box was a circular piece of cardboard that last night's dinner had once rested on.

Using 10th grade geometry, he drew two cords on the circular piece of cardboard to find the center of the circle. And using the straight edge of the pizza box, he drew a diameter on the cardboard circle. You with me so far?

TOM: I'm with you!

RAY: Using one of the 90-degree corners of the box, he drew a radius perpendicular to that diameter. Circle, diameter, and a radius, okay? Carefully with his jackknife, he cut the circle in half and discarded the piece without the radius drawn on it.

TOM: So he's now got half of the circle.

RAY: Exactly. He's got a semi-circle with a radius drawn on it. Now, he takes that remaining piece and stands it up, kind of like a slice of watermelon, so that radius is perpendicular to the table top. This piece of cardboard represents a cross-section of the fuel in the lower half of the fuel tank.

Now somewhere along the radius line that's perpendicular to the table is the point that represents the quarter-full mark that we're looking for. We know it's not the midpoint, because the semi-circle is smaller at the bottom than at the top.

The truck driver takes his pencil in one hand, with the sharpened end pointing up toward the sky, and he uses trial and error to find the point along the radius that will balance the piece of cardboard, like you would balance a dinner plate on a fingertip. When he does that, he's found the center of mass of that piece of cardboard.


TOM: Get out!

RAY: And if it's the center of mass of that piece of cardboard, it's also got to be the center of mass of the gasoline that's in the tank. So half the fuel has to be below that mark, and half the fuel has got to be above it. In other words, he's found the quarter-full mark by finding the center of mass of that piece of cardboard.

In the original puzzler, the fellow said the tank was 20-inches in diameter. But, it doesn't make any difference what the diameter of the tank is, since all circles are the same. If you find the center of mass of one circle, or semi-circle in this case, you've found it for all of them.
It turns out to be almost exactly 40% of the distance from the center of the circle. That's where the pencil would balance the cardboard.

TOM: Wow. Pretty neat!

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