(Solved):   The diagram shows a right triangular prism of height \( h \) inside a right pyramid. The p ...

 

The diagram shows a right triangular prism of height \( h \) inside a right pyramid. The pyramid has a height of 12 and a base that is an equilateral triangle, \( A B C \), of side 8 . The base of the prism sits on the base of the pyramid. Points \( P, Q \) and \( R \) lie on the edges \( O A, O B \) and \( O C \), respectively, of the pyramid \( O A B C \). Pyramids \( O A B C \) and \( O P Q R \) are similar. (a) Show that the volume, \( V \), of the triangular prism is given by \( V=\frac{\sqrt{3}}{9}\left(a h^{3}+b h^{2}+c h\right) \) where \( a \), \( b \) and \( c \) are integers to be found.