So I heard an interesting problem on the radio, can't crack it. A trucker's gas gauge is broken, he knows that the tank diameter is 20 inches, if he dips the dipstick and gas is at half the length obviously it's half-full. What height does he need to mark off on the dipstick in order to measure when the tank is 1/4 full?

I figure let's forget about the unnecessary third dimension and solve this for a unit circle. I tried thinking of this as a circle where I'm solving for the top 1/4 area and I don't know the length from the "fluid level" to the top of the tank; nor the length from the

line to the point where the fluid level meets the edge of the tank. So I try setting up my function as

where

is the necessary number to set the fluid-level at the x-axis. I then need to integrate from 0 to

where

is the length of the fluid-level line from the

line to the edge of the tank. Integrating

and setting it equal to

is a bit laborious and gives an answer that doesn't really seem to help.

I've heard that there is a purely trigonometric solution, but I don't know any trigonometry that deals with area except the basic formula

.