The Semicylindrical Trough

In view of the last two analyses that we have done, geometry is the only question posed by this problem. We need the relationship between r[t] and h[t] once again. One line in the diagram holds the key to this. That line is the hypotenuse of the triangle which has r[t] for one leg and R — h[t] for the other. The Pythagorean theorem links the three  

The first solution is unphysical, so we substitute the second into the expression for dV[t]  

Replacing this in the naaterial balance (assuming all densities are constant and equal everywhere), we find  

The solution of this equation involves an integral on the left-hand side that results in an implicit solution for h[ t ]. We can try to solve directly for h[ t ], but Mathematica cannot do it  

The equations appear to involve the variables to be solved for in an essentially non-algebraic way. 

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A variation of the left-hand side theme and the issues of geometry is that of the semicylindrical trough lying on its side. The physical situation is quite similar to that of the triangular trough, except that the walls follow a circular curve. The physical system is shown in Figure 6. 

This is similar to the equation that we encountered in the last problem for the semicylindrical trough ... 

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