301. (15 pts) Your company has purchased intellectual property to a newly discovered laboratory process in which the following chemical reactions
convert raw materials A and B to a valuable product T:
A + B \rightarrow X + Y
B + X \rightarrow 2Z
Y + Z \rightarrow T
Among the compounds in this reaction mechanism, only A and B are available in the market (at $10/mol each), whereas X, Y and Z are not available for
purchase. There is a big market for T in the United States (current selling price $50/mol), but little demand for X, Y or Z. Exporting X, Y or Z to other
countries is not an option at this point.
To leverage this new IP acquisition, your company needs to design a chemical process that minimizes the cost of feed to produce one mole of T.
A. In order to maximize profit, could you design a process to produce T by avoiding net production of all three intermediates X, Y and Z?
B. Calculate the minimum number of moles of feed and the molar proportion of the feed required by the process to produce 1 mol T. Mathematically, one
approach is to assume $x_1$ and $x_2$ as the coefficients of two of the chemical reactions and minimize a quantity related to $x_1$ and $x_2$. Another slightly longer
approach is to assume $x_1$ and $x_2$ to be the moles of A and B in the feed, $x_3$ and $x_4$ to be the coefficients of two of the reactions, and to minimize the
quantity $2x_1 + x_2$ (or maximize $-[2x_1 + x_2]$). Report the number of moles and molar composition of this optimal feed. You must solve this problem
graphically.
Note: It is a bit awkward to employ variables such as $x_1$ to represent number of moles, but let us stick to the standard LP notation where all variables are
represented by $x_i$.
