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Marketing Research (Cost, Revenue, Etc.)

1. Homework Statement
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

336 $1.50
315 $2.00
265 $2.50
242 $3.00
208 $3.50
176 $4.00
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
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?

2. Homework Equations


3. 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.

2. R(x)=px

3. C(x)=47.3575+(x*1.182)

The domain is 0<x<400

4. P(x)=$3.50x-[47.3575+(x*1.182)]

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!
 
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Ray Vickson

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1. Homework Statement
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

336 $1.50
315 $2.00
265 $2.50
242 $3.00
208 $3.50
176 $4.00
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
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?

2. Homework Equations


3. 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.

2. R(x)=px

3. C(x)=47.3575+(x*1.182)

The domain is 0<x<400

4. P(x)=$3.50x-[47.3575+(x*1.182)]

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!
(i) Your profit equation is wrong: where did you get the term 3.50 x? The hint told you what you need to do.

(ii) To maximize the profit, you need to set its derivative equal to zero. That makes this a bit of a calculus problem, rather than a pre-calculus one, although in this case the problem is also solvable without calculus.

(iii) Where did the figures of 208 and 176 come from in 5) and 6)?

(iv) I use your demand equation P = 6.65 - 0.015 x I get a maximum profit of $450.96/week, by producing 182 loaves per week. If I use a more exact demand equation (obtained by keeping more significant figures) in the least-squares fit of price vs number, I get a maximum profit of $415.40/week, by producing 175 loaves per week. If I use your solution of x = 182 in the more exact profit equation I get $414.58/week.
 
(i) Your profit equation is wrong: where did you get the term 3.50 x? The hint told you what you need to do.

(ii) To maximize the profit, you need to set its derivative equal to zero. That makes this a bit of a calculus problem, rather than a pre-calculus one, although in this case the problem is also solvable without calculus.

(iii) Where did the figures of 208 and 176 come from in 5) and 6)?

(iv) I use your demand equation P = 6.65 - 0.015 x I get a maximum profit of $450.96/week, by producing 182 loaves per week. If I use a more exact demand equation (obtained by keeping more significant figures) in the least-squares fit of price vs number, I get a maximum profit of $415.40/week, by producing 175 loaves per week. If I use your solution of x = 182 in the more exact profit equation I get $414.58/week.
I wasn't sure if I was just supposed to draw from the amounts that had previously been sold. I just took 3.50 from how much it had been sold for (from those listed prices) that gave maximum revenue. What am I supposed to use for the profit equation?
 
Based on (1), you know the relation between p and x and should use that here.

A month does not have exactly 4 weeks.

Why did you use $3.5 for the profit function?

How do you find the maximum of a function in general?
What should I use instead of the 4 weeks then? And I used 3.5 because of the prices that I sold for in the past, it gave the most revenue. And I don't know how to find the max of a function in general. Above poster said set the derivative to 0, but what would that look like?
 

Ray Vickson

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Homework Helper
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I wasn't sure if I was just supposed to draw from the amounts that had previously been sold. I just took 3.50 from how much it had been sold for (from those listed prices) that gave maximum revenue. What am I supposed to use for the profit equation?
Let me repeat: read the Hint that goes along with question 4.

However, first you need to finish question 2. You wrote R(x) = px, but that is not yet a usable answer because you have not said how to find (or to express) p in terms of x. As it stands, you have R(x,p), not R(x).
 
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Let me repeat: read the Hint that goes along with question 4.
So for the profit function do I just put it as P(x)=px-(47.3575+1.182x) and leave it as that?
 

Ray Vickson

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No, no, no! Tell us what 'p' is!
And how do I find that? I'm sorry, I realize this is probably more frustrating for you than me, but math is not my strong suit (obviously).
 
No, no, no! Tell us what 'p' is!
The reason I said px is because p was used as the variable for price above. So I had profit=price*number sold - cost
 

Ray Vickson

Science Advisor
Homework Helper
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The reason I said px is because p was used as the variable for price above. So I had profit=price*number sold - cost
What prevents me from choosing x = 200 and p = $10,000,000? That would give me a nice profit of almost $2 billion per week. Obviously, I cannot do that, but the question is why not? Something is stopping me. What is it?
 
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