Putnam 1980 B2

Let $S$ be the solid in three-dimensional space consisting of all points $(x,y,z)$ satisfying the following six simultaneous conditions: $$x,y,z \geq 0, \; x+y+z\leq 11, \; 2x+4y+3z \leq 36, \; 2x+3z \leq 44.$$ a) Determine the number $V$ of vertices of $S.$ b) Determine the number $E$ of edges of $S.$ c) Sketch in the $bc$ -plane the set of points $(b, c)$ such that $(2,5,4)$ is one of the points $(x, y, z)$ at which the linear function $bx + cy + z$ assumes its maximum value on $S.$