SOLUTION: A local grocery store has agreed to sell your homemade bread. You will use the following information along with some ideas from Chapter 3 to decide how many loaves should be manufa

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Question 715304: A local grocery store has agreed to sell your homemade bread. You will use the following information along with some ideas from Chapter 3 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 5.7.) 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

1. 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?
 
2. 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.)
 
3. 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?
 
4. Profit Function. Find P(x), the weekly profit for producing and selling x loaves of bread. (Hint: profit = revenue – cost.)
 
5. 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?
 
6. 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?
 
7. Conclusion. How many loaves of bread will you produce each week and how much will you charge for each loaf? Why?
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