We know the shape of the cap and we know the area of the base, and the volume so we can find the unique height, , unique because the volume will be some multiple of the height times the area of the base. If we know the height we can find the radius, of the ball with a cap with dimensions and find the angle with a little geometry as in this diagram.
Personally I want to stop here. The interesting part is over. Yet the problem is to actually find so let’s press on. From the diagram, So it remains to find the relationship between We could look up the formula. We could use a CAS (computer algebra system), or we could derive it. I will use the CAS when I really need it, so for now let’s find a formula for the volume of a cap given the radius of the ball and the radius of the cap. I will use the calculus method of volume by disks with the setup as in this diagram.
We form the cap by rotating the yellow section about the y-axis. Think of it as composed of disks formed by rotating the blue-shaded rectangle about the y-axis. Solving the equation of the semicircle for the volume integral is
So now we have an equation that relates Given and we could use numerical techniques to get or we could attempt to solve for by isolating the square root, squaring both sides to arrive at a sixth degree equation. I give. Time for a CAS. This is what Wolfram Alpha gives for the solution.
I have omitted the two other solutions which were complex number expressions since we are looking for a real number answer. Briefly I wonder if there isn’t a more elegant way to find an expression for that gets directly to the real number. Anyway we now have an expression for in terms of and and I’ve had it. Nothing particularly beautiful or surprising here.
Wait. We wanted to know the angle . This is a practical problem. All those pesky engineering questions intrude. We need to check if the formula is correct by using Wolfram Alpha or preferably another CAS. We need to check if typical values for and give the anticipated values for . We also need to know the sensitivity of to the uncertainty of the measurements for and . This is engineering – turning a math model into a real-life solutions generator, tedious and often ill-conditioned requiring patience and attention to detail – not my strongest trait. That’s why I would make a crummy engineer.