** 1. ** | The Hickory Cabinet and Furniture Company makes chairs. The fixed cost per month of making chairs is $7,500, and the variable cost per chair is $40. Price is related to demand according to the following linear equation. v = 400 1.2 p Develop the nonlinear profit function for this company and determine the price that will maximize profit, the optimal volume, and the maximum profit per month. |

** 2. ** | Graphically illustrate the profit curve developed in problem 1. Indicate the optimal price and the maximum profit per month. |

** 3. ** | The Rainwater Brewery produces beer. The annual fixed cost is $150,000, and the variable cost per barrel is $16. Price is related to demand according to the following linear equation. v = 75,000 1,153.8 p Develop the nonlinear profit function for the brewery and determine the price that will maximize profit, the optimal volume, and the maximum profit per year. |

** 4. ** | The Rolling Creek Textile Mill makes denim. The monthly fixed cost is $8,000, and the variable cost per yard of denim is $0.35. Price is related to demand according to the following linear equation. v = 17,000 5,666 p Develop the nonlinear profit function for the textile mill and determine the optimal price, the optimal volume, and the maximum profit per month. |

** 5. ** | The Grady Tire Company recaps tires. The weekly fixed cost is $2,500, and the variable cost per tire is $9. Price is related to demand according to the following linear equation. v = 200 4.75 p Develop the nonlinear profit function for the tire company and determine the optimal price, the optimal volume, and the maximum profit per week. |

** 6. ** | Andy Mendoza makes handcrafted dolls, which he sells at craft fairs. He is considering massproducing the dolls to sell in stores. He estimates that the initial investment for plant and equipment will be $25,000, while labor, material, packaging, and shipping will be about $10 per doll. He has determined that sales volume is related to price according to the following linear equation. v = 4,000 80 p Develop the nonlinear profit function for Andy and determine the price that will maximize profit, the optimal volume, and the maximum profit per month. |

** 7. ** | The Rainwater Brewery produces beer, which it sells to distributors in barrels. The brewery incurs a monthly fixed cost of $12,000, and the variable cost per barrel is $17. The brewery has developed the following profit function and demand constraint. maximize Z = vp $12,000 17 v subject to v = 800 15 p [Page D-8] Solve this nonlinear programming model for the optimal price ( p ) using the substitution method. |

** 8. ** | The Beaver Creek Pottery Company has developed the following nonlinear programming model to determine the optimal number of bowls ( x _{ 1 } ) and mugs ( x _{ 2 } ) to produce each day. maximize Z = $7 x _{ 1 } 0.3 x ^{ 2 } _{ 1 } + 8 x _{ 2 } 0.4 x ^{ 2 } _{ 2 } subject to 4 x _{ 1 } + 5 x _{ 2 } = 100 hr Determine the optimal solution to this nonlinear programming model using the substitution method. |

** 9. ** | The Evergreen Fertilizer Company produces two types of fertilizers, Fastgro and Super Two. The company has developed the following nonlinear programming model to determine the optimal number of bags of Fastgro ( x _{ 1 } ) and Super Two ( x _{ 2 } ) that they must produce each day to maximize profit, given a constraint for available potassium. maximize Z = $30 x _{ 1 } 2 x ^{ 2 } _{ 1 } + 25 x _{ 2 } 0.5 x ^{ 2 } _{ 2 } subject to 3 x _{ 1 } + 6 x _{ 2 } = 300 lb Determine the optimal solution to this nonlinear programming model using the substitution method. |

** 10. ** | The Rolling Creek Textile Mill produces denim and brushed-cotton cloth. The company has developed the following nonlinear programming model to determine the optimal number of yards of denim ( x _{ 1 } ) and brushed cotton ( x _{ 2 } ) to produce each day to maximize profit, subject to a labor constraint. maximize Z = $10 x _{ 1 } 0.02 x ^{ 2 } _{ 1 } + 12 x _{ 2 } 0.03 x ^{ 2 } _{ 2 } subject to 0.2 x _{ 1 } + 0.1 x _{ 2 } = 40 hr Determine the optimal solution to this nonlinear programming model using the substitution method. |

** 11. ** | Solve problem 8 using the method of Lagrange multipliers. |

** 12. ** | Solve problem 9 using the method of Lagrange multipliers. |

** 13. ** | Solve problem 10 using the method of Lagrange multipliers. |

** 14. ** | The Riverwood Paneling Company makes two kinds of wood paneling, Colonial and Western. The company has developed the following nonlinear programming model to determine the optimal number of sheets of Colonial paneling ( x _{ 1 } ) and Western paneling ( x _{ 2 } ) to produce to maximize profit, subject to a labor constraint. maximize Z = $25 x _{ 1 } 0.08 x ^{ 2 } _{ 1 } + 30 x _{ 2 } 1.2 x ^{ 2 } _{ 2 } subject to x _{ 1 } + 2 x _{ 2 } = 40 hr [Page D-9] Determine the optimal solution to this nonlinear programming model using the method of Lagrange multipliers. |

** 15. ** | Interpret the meaning of l , the Lagrange multiplier , in problem 14. |