You are given the final simplex tableau for the dual problem. Give the solution to the primal problem and the solution to the associated dual problem. Problem: Minimize \(\quad C=2 x+3 y\) subject to $\begin{aligned} x+4 y & \geq 8 \\ x+y & \geq 5 \\ 2 x+y & \geq 7 \\\ x \geq 0, y & \geq 0 \end{aligned}$ Final tablcau: $$ \begin{array}{cccccc|c} u & v & w & x & y & P & \text { Constant } \\ \hline 0 & 1 & \frac{7}{3} & \frac{4}{3} & -\frac{1}{3} & 0 & \frac{5}{5} \\ 1 & 0 & -\frac{1}{3} & -\frac{1}{3} & \frac{1}{3} & 0 & \frac{1}{3} \\ \hline 0 & 0 & 2 & 4 & 1 & 1 & 11 \end{array} $$

Use the method of this section to solve each linear programming problem. $$ \begin{aligned} \text { Minimize } & C=-2 x+3 y \\ \text { subject to } & x+3 y \leq 60 \\ 2 x+y & \geq 45 \\ x & \leq 40 \\ x \geq 0, y & \geq 0 \end{aligned} $$

Solve each linear programming problem by the simplex method. $$ \begin{array}{ll} \text { Maximize } & P=5 x+4 y \\ \text { subject to } & 3 x+5 y \leq 78 \\ & 4 x+y \leq 36 \\ & x \geq 0, y \geq 0 \end{array} $$

You are given the final simplex tableau for the dual problem. Give the solution to the primal problem and the solution to the associated dual problem. Problem: Minimize \(C=3 x+2 y\) subject to $\begin{aligned} 5 x+y & \geq 10 \\ 2 x+2 y & \geq 12 \\ x+4 y & \geq 12 \\ x \geq 0, y & \geq 0 \end{aligned}$ Final tablcau: $$ \begin{array}{rrrrrr|r} u & v & w & x & y & P & \text { Constant } \\ \hline 1 & 0 & -\frac{1}{4} & \frac{1}{4} & -\frac{1}{4} & 0 & \frac{1}{4} \\ 0 & 1 & \frac{19}{8} & -\frac{1}{8} & \frac{5}{8} & 0 & \frac{7}{8} \\ \hline 0 & 0 & 9 & 1 & 5 & 1 & 13 \end{array} $$

Solve each linear programming problem by the simplex method. $$ \begin{array}{ll} \text { Maximize } & P=3 x+4 y+z \\ \text { subject to } & 3 x+10 y+5 z \leq 120 \\ & 5 x+2 y+8 z \leq 6 \\ & 8 x+10 y+3 z \leq 105 \\ & x \geq 0, y \geq 0, z \geq 0 \end{array} $$

A veterinarian has been asked to prepare a diet for a group of dogs to be used in a nutrition study at the School of Animal Science. It has been stipulated that each serving should be no larger than \(8 \mathrm{oz}\) and must contain at least 29 units of nutrient \(I\) and 20 units of nutrient II. The vet has decided that the diet may be prepared from two brands of dog food: brand \(\mathrm{A}\) and brand \(\mathrm{B}\). Each ounce of brand A contains 3 units of nutrient \(\mathrm{I}\) and 4 units of nutrient II. Each ounce of brand \(B\) contains 5 units of nutrient I and 2 units of nutrient II. Brand \(A\) costs 3 cents/ounce and brand B costs 4 cents/ounce. Determine how many ounces of each brand of dog food should be used per serving to meet the given requirements at minimum cost.