A Truck, a Box, and More Than 8th grade Mathematics!
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!
[ Car Talk Puzzler ]