The following table shows how many hours in process reactors A and B are required to produce 1 ton each of chemical products 1, 2, and 3. The two reactors are available for 35 and 40 hours per week, respectively.

Let x, y, and z be the number of tons each of products 1, 2, and 3 that can be produced in one week.

a. Use the data in the table to write two equations in terms of x, y, and z. Determine whether a unique solution exists. If not, use MATLAB to find the relations between x, y, and z.

b. Note that negative values x, y, and z have no meaning here. Find the allowable ranges for x, y, and z.

c. Suppose the profits for each product are $200, $300, and $100 for products 1, 2, and 3, respectively. Find the values of x, y, and z to maximize the profit.

d. Suppose the profits for each product are $200, $500, and $100 for products 1, 2, and 3, respectively. Find the values of x, y, and z to maximize the profit.