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
In order to increase manufacturing capacity, you’ve taken out a loan to buy an industrial sized oven for $3800. 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 9% compounded monthly. The monthly payments are $189.43. (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
- 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?
The Attempt at a Solution
I have my scatter plot created with a line of best fit, but I'm unsure of how to get the equation for that line from excel. Here are the answers I have so far:
1. Yes, the relationship appears to be linear.
The domain is 0<x<400
5.The maximum revenue comes from selling 208 loaves at $3.50 a loaf, which nets $728.00 in revenue.
6. The maximum profit comes from selling 176 loaves at $4.00 a loaf, which nets $448.61 in profit.
7. I will produce 176 loaves each week and will sell them for $4.00 a loaf. The reasoning behind this is that it will get me the most profit. While it doesn’t bring in as much revenue, the difference in cost is enough to make it more worthwhile.
I really want to understand this but am having trouble figuring out the maximum parts. Do I just graph the data? I appreciate your help.
Update: I got the demand equation as P=6.65-.015x
Update2: I think I figured out the rest of the equation but if you could double check my work I would really appreciate it. Thank you!