Ah! That’s a good practical solution if you don’t want to get out your graphing calculator in the rest stop and start solving trancendental equations! LOL.
It looks like this has been sorted out. You’re right, the answer is incorrect. By definition, the center of mass is the point about which MOMENTS are balanced (as DJohnC pointed out). That’s why you can balance the half-circle with a pen at the CM. This does NOT mean that masses are balanced (as your seesaw example demonstrates.)
Center of mass is important because it’s the point at which the application of a force causes purely translational motion (no rotation).
Thanks to those who submitted a derivation. I won’t add mine. If you use a triangle instead of a semicircle, you can prove it pretty quickly by geometry.
The Center of Mass solution is certainly incorrect for all the reason noted above.
Here is a 6th grade solution that is a good approximation to the exact, noting that it’s easier to approximate the height of the “third” quarter tank instead of the last one.
First approximation:
Because the walls of the tank are fairly close to vertical near the mid-line we can assume the second quarter from the bottom to be a near rectangle. A 20" dia. tank has a 314 in.sq. cross section. One quarter of that is 78.5 in.sq. Divide that by the 20" width and get 3.925" from the mid-line.
Second approximation:
Using Pythagoras calculate width at the liquid surface getting 18.4 in. Now assume the liquid cross section to be a trapezoid instead of a rectangle and divide the 78.5 in.sq. by 19.2" (the average of the 20" width at the top and 18.4" at the bottom). The result is 4.09".
Third approximation:
Repeat above calculation but use the tank width at “half depth” (2in. from mid-line) or 19.6" for the mid-line width of the trapezoid. This nets you 4.005".
Fourth approximation:
Take the average of the second and the third. At 4.046" we are three iterations beyond splitting hairs.
“The calculus solutions are excellent for showing that you are starting with two different equations, so it is not surprising to get two different answers.”
I can give you an arbitrary number of different answers. It would be Nobel Prize winningly surprising to get two different CORRECT answers.
That could be right, I haven’t bothered to check your math, just assume it’s right.
Since we are now accepting approximations on our pretend radio show:
Here’s an easier approximation, one that requires NO math.
When the tank’s half full, that’s about a quarter full. Time to fill up.
I believe that the correct laser solution is to use a surgical laser instead of a jigsaw.
While the example given on the show may not be 100% correct down to the ounce, I think it’s safe to say that for the purpose given in the problem, (finding when he needs to refuel), it is close enough. (even for government work ) 
It is close enough to the correct answer, but you hardly know that without knowing the correct answer. Letting the fuel line drop to 5 inches from the bottom of a 20-inch diameter tank – half way to the bottom from half full – is also close for the purpose given in the problem, but I think that answer was rejected when the puzzler was given. Using the centroid, on the other hand, was touted for its (bogus) scientific excellence.
“Letting the fuel line drop to 5 inches from the bottom of a 20-inch diameter tank – half way to the bottom from half full – is also close for the purpose given in the problem”
Your right, and is probably as accurate as most fuel gauges…whose accuracy (or lack there of) have also been covered on the site.