I was helping my 17 year old daughter (just starting calculus) with the optimization problem of maximizing the volume of a right circular cone that can inscribed in a sphere. She tried what she thought was a short cut by using a cone with vertex at the center the sphere (instead of the top) and couldn’t understand why it didn’t yield the right answer. I tried to explain that she solved a different problem but she couldn’t understand why the solution to the simpler problem wasn’t also the solution to the stated problem. It didn’t help that the very next problem was a distance problem where the book suggested a short cut of minimizing the square of the distance rather than the distance (to avoid square roots). To her, both were the same (i.e. a logical shortcut). I’ve since been struggling with how to explain the apparent discrepancy. I’ve thought about using a triangle/circle analogy and say that the two triangles aren’t similar. Any other ideas?