A furniture manufacturer makes two types of furniture - chairs and sofas. The production of the sofas and chairs requires three operations - carpentry, finishing, and upholstery. Manufacturing a chair requires 3 hours of carpentry, 9 hours of finishing, and 2 hours of upholstery. Manufacturing a sofa requires 2 hours of carpentry, 4 hours of finishing, and 10 hours of upholstery. The factory has allocated at most 66 labor hors for carpentry, 180 labor hours for finishing, and 200 labor hours for upholstery. The profit per chair is $90 and the profit per sofa is $75. Formulate an optimization problem for the solution of which gives the numbers of chairs and sofas that should be produced each day to maximize profit. DO NOT SOLVE the resulting optimization problem! At least 30 pounds of deluxe coffee is to be mixed with regular coffee to make at least 90 pounds of blended coffee. Deluxe and regular coffees cost $7 and $3 per pound, respectively. Formulate an optimization problem for the solution of which gives the numbers of pounds of deluxe and regular coffees that should be used to minimize cost. DO NOT SOLVE the resulting optimization problem! A developer is planning to build a new subdivision consisting of townhouses, single-story detached houses and two-story detached houses. On one acre he can put 6 townhouses or 4 single-story houses or 2 two-story-houses and he has 60 acres available. It costs him $40,000 to build each townhouse, $50,000 to build each single-story house and $60,000 to build each two-story house. He makes a profit of $15,000 on each townhouse, $18,000 on cach single-story house and $20,000 on each two-story house and he has $2,880,000 of capital available. Townhouses require 2500 hours of labor, single-story houses require 3000 hours of labor and two-story houses require 4000 hours of labor and he has 240,000 hours of labor available. Formulate an optimization problem for the solution of which gives the numbers of houses of each type that he should construct in order to maximize his profit. DO NOT SOLVE the resulting optimization problem!