Dear Tom and Ray,
I don’t really listen to your show. Our cats do many things to annoy us, and one of them is walking on the “on” switch on the clock radio while your show is on.
So I heard the “solution” to the cylindrical gas tank puzzle. I’m afraid I have to differ with you. The center of mass need NOT have half the mass on one side and half on the other: it also matters how far from the center the mass is. A 200 pound man (no offense) sitting three feet from the middle of a see-saw will balance his 120 pound significant other sitting five feet from the middle; but 5/8 of the mass is on one side, and 3/8 on the other.
The center of mass of the 20" diameter half disk is at a distance of 40/(3 pi) from the center, or about 4.244 inches.
To see how much area is below this point is a matter of 11th grade trigonometry. On your pizza half disc, draw two radii from the center to the chord through the center of mass (the gas level). Those radii make an angle theta with the top of the disc; you know that sin(theta)=4/(3 pi), so you can figure out that theta (in radians) must be arcsin(4/(3 pi)) = .4383137. So the chords make a big slice of pie, whose area is
100((pi/2) - theta) = 113.248.
That’s the piece below the center of mass, EXCEPT that it has on top of it an extra triangle; so you have to subtract the area of the triangle. Its height is 10sin(theta) = 4.383137, and its base is 20cos(theta)=18.10937. So the area of the triangle is half the base times the height, or 39.687. So the area below your gas line is 113.248 - 39.687 = 73.561.
The total area of the full pizza disk is 100pi = 314.159, so your method marks the point where the tank is at
73.561/314.159 = .23415:
barely 23.4% full!
You may say that this is, as the saying goes, “good enough for government work.” vehicle which is supposed to arrive with the 25% fuel load needed to return, when in fact there is only 23.4%. If somebody has to get out at the moon and walk, it won’t be me.
Member, Department of Mathematics
Your Alma Mater
P.S. Actually finding the right place for the quarter-tank mark involves solving some transcendental equation that I don’t know how to type. Doing this numerically is a fine exercise for Newton’s method, which I assume that you enjoyed in MIT course 18.01, long, long ago in a galaxy far, far away.