Brimful

Can you find the volumes of the mathematical vessels?

Problem

 

Constants $A, B, C, D$ are chosen so that the following $4$ curves pass through the point $(2.5, 10)$ $$y = Ax\quad y = Bx^2 \quad y = Cx^3\quad y = Dx^4+x$$

What values must the constants take? Can you identify each curve in the following accurately drawn chart?

 

Image
Brimful



These curves are now used to design some mathematical vessels of height $10$ by rotating the curves about the $y$ axis. Assuming that $x$ and $y$ are measured in centimetres, what are the volumes of the vessels?

Water is poured slowly at a rate of 1cm$^3$ per minute into these vessels. At what depth of water, to the nearest mm, will each of them be half full? Do your results make sense from the diagram?