Chapter 5.3, Problem 57E

In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method.

Pollution Control. Because of new federal regulations on pollution, a chemical plant introduced a new, more expensive process to supplement or replace an older process used in the production of a particular chemical. The older process emitted $20$ grams of sulfur dioxide and $40$ grams of particulate matter into the atmosphere for each gallon of chemical produced. The new process emits $5$ grams of sulfur dioxide and $20$ grams of particulate matter for each gallon produced. The company makes a profit of $60\cancel{\text{c}}$ per gallon and $20\cancel{\text{c}}$ per gallon on the old and new process, respectively.

(A) If the government allows the plant to emit no more than $16,000$ grams of sulfur dioxide and $30,000$ grams of particulate matter daily, how many gallons of the chemical should be produced by each process to maximize daily profit? What is the maximum daily profit?

(B) Discuss the effect on the production schedule and the maximum profit if the government decides to restrict emissions of sulfur dioxide to $11,500$ grams daily and all other data remain unchanged.

(C) Discuss the effect on the production schedule and the maximum profit if the government decides to restrict emissions of sulfur dioxide to $7,200$ grams daily and all other data remain unchanged.