**Market Research Project**

A local grocery store has agreed to sell your homemade bread. You will use the following information along with some ideas from Chapter 1 to decide how many loaves should be manufactured each week and what price should be charged.

After tracking weekly sales at several different prices, you get the following data:

Loaves Sold, x |
Price, p |

355 | $1.50 |

320 | $2.00 |

265 | $2.50 |

235 | $3.00 |

180 | $3.50 |

125 | $4.00 |

In order to increase manufacturing capacity, you’ve taken out a loan to buy an industrial sized oven for $4000. The new oven will allow you to make a maximum of about 400 loaves of bread per week. The loan is to be paid back monthly over two years at an annual interest rate of 10% compounded monthly. The monthly payments are $203.40. (You can check these numbers after section 2.2.) The ingredients for two loaves of bread are given in the table below. The $1.182 is the cost of the ingredients for a single loaf of bread.

ingredients |
price/package size |
price / single loaf |

5 cups flour | $3.86 / 19 cups | $0.508 |

3 Tbs. sugar | $4.98 / 378 Tbs. | $0.020 |

2 tsp. salt | $0.52 / 122 ¾ tsp. | $0.004 |

¼ tsp. baking soda | $0.60 / 100 ¾ tsp. | $0.001 |

1 package dry yeast | $0.66 / package | $0.330 |

1 cup buttermilk | $1.17 / 4 cups | $0.146 |

1/3 cup milk | $2.38 / gallon | $0.025 |

1 egg | $2.35 / dozen | $0.098 |

packaging | $0.050 | |

Total |
$1.182 |

**Demand Equation.**Make a scatter plot of the six data points (using the number sold as the x-coordinate.) Does the relationship appear to be linear? Use regression analysis to find the line of best fit. This line will be your demand equation. How strong is the correlation?

**Revenue Function.**Find*R*(*x*), the weekly revenue as a function of loaves sold,*x.*(Note that*R*(*x*) is an equation not a single value.)

**Cost Function.**Find*C*(*x*), the weekly cost for producing*x*loaves of bread. Be sure to include both the cost of the oven and the ingredients. What is the domain of the cost function?

**Profit Function.**Find*P*(*x*), the weekly profit for producing and selling*x*loaves of bread. (Hint: profit = revenue – cost.)

**Maximum Revenue.**Find the number of loaves that should be sold in order to maximize revenue. What is the maximum revenue? What price should be charged in order to maximize revenue?

**Maximum Profit.**Find the number of loaves that should be produced and sold in order to maximize the profit. What is the maximum profit? What price should be used to maximize profit?

**Conclusion.**How many loaves of bread will you produce each week and how much will you charge for each loaf? Why?