We have a perfect cylinder with a diameter of 3 ft that lies horizontal.
The gas gauge is broken so we are forced to use a dipstick to determine how much gas in our tank. In this problem we are trying to figure out at what point on the "dipstick" corresponds to our tank being 3/8 full.
I'd like to do this using a rootfinding method, any ideas.
This is a variant of the well-known "quarter-tank" problem. You can solve it by reminding that the area of a circular segment of radius $r$ and height $h$ is
$$r^2 \cos^{-1} (\frac{r-h}{r}) - (r-h) \sqrt{2rh-h^2}$$
Then, we can find the level corresponding to being $3/8$ full by solving
$$\cos^{-1} (1-x) - (1-x) \sqrt{2x-x^2}=\frac {3}{8}\pi$$
where $x=h/r$.
This equation cannot be solved analytically, just as the similar equation appearing in the quarter-tank problem (which is identical except for the presence of $\frac{\pi}{4}$ in the RHS). However, the solution can be found numerically. Its value is $r \cdot 0.8023560468....$.