Actually, what I gave is what we in the bizness call an "answer," not a "solution." A solution would say something like: Assume the circle has radius 1. We are looking for a chord inside the circle such that the chord divides the circle into a piece with area pi/4 and a piece with area 3pi/4. Suppose the chord subtends an angle t. The area of the smaller piece is that of the "pie wedge" containing the chord, minus that of the triangle whose sides are the chord and the two radii drawn to the ends of the chord. The former has area t/2 and the latter has area 1/2*sin t. t thus satisfies t - sin t = pi/2. The distance from the chord to the center is sin t/2, so the distance to the edge is 1 - sin t/2. Scaling by the radius r gives the answer.

This would be easier if there were a picture.

Noah's estimate with the correct number on the right hand side gives t = 2.11, and 2.11 - sin 2.11 = 1.25, whereas pi/2 = 1.57. Not so hot. Presumably the guy wants to know where the quarter mark is on the tank so he can refill it when it gets there. If he's willing to live life on the edge we could give him a better estimate. Noah's is for 1/8; for 1/16 it gives .36 vs. .39, pretty good.

The actual answer is t = 2.30988. Time for dinner.