Question

The water level in a hemispherical bowl of radius 12 inches is 4.6 inches. What angle can you tilt the bowl before the water begins to spill?

Answer 1

You can tilt the bowl by `38.1°` before the water spills.

Explanation:

On the image above, you can see the bowl with water as prensented in the problem and a hypothetical tilted bowl with the water reaching the bowl's edge. The two hemispheres centers are superimposed and the two diameters form an angle a.

The same angle is found in the right triangle formed with:
-the segment from the hemisphere's center to the water surface's center (`12-4.6=7.4` inches)
-the segment from the hemisphere's center to the water surface's edge (`12` inches)
-the segment from the water surface's center to its edge

In this triangle, `sin(a)=7.4/12`

therefore `a=sin^(-1)(7.4/12)~~38.1°`