# Marketing Research (Cost, Revenue, Etc.)

• frownifdown
In summary, a local grocery store has agreed to sell homemade bread and the goal is to determine the optimal number of loaves to produce and the price to charge. After tracking weekly sales at different prices, a scatter plot is created and a linear demand equation is found using regression analysis. The correlation appears to be strong. The revenue function is found to be R(x)=px, the cost function is C(x)=47.3575+(x*1.182), and the profit function is P(x)=6.65-.015x. The maximum revenue is achieved by selling 208 loaves at $3.50 per loaf, while the maximum profit is achieved by selling 176 loaves at$4.00 per loaf. The
frownifdown

## 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? ## Homework Equations ## 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! Last edited: frownifdown said: 2. R(x)=px Based on (1), you know the relation between p and x and should use that here. frownifdown said: 3. C(x)=47.3575+(x*1.182) A month does not have exactly 4 weeks. Why did you use$3.5 for the profit function?

frownifdown said:
5. I am not sure how to figure out maximum revenue
How do you find the maximum of a function in general?

frownifdown said:

## 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? ## Homework Equations ## 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.

Ray Vickson said:
(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? mfb said: 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?

frownifdown said:
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).

Last edited:
Ray Vickson said:
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?

frownifdown said:
So for the profit function do I just put it as P(x)=px-(47.3575+1.182x) and leave it as that?

No, no, no! Tell us what 'p' is!

Ray Vickson said:
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).

Ray Vickson said:
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

frownifdown said:
And how do I find that?
See post 2, first part. The necessary hints are all in the thread now I think, you just have to follow them.

frownifdown said:
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?

Last edited:

## What is marketing research?

Marketing research is the process of gathering and analyzing data about the market, customers, and competitors in order to make informed business decisions. It involves using various methods and techniques to collect and interpret data related to a company's products, services, and target audience.

## Why is marketing research important?

Marketing research is important because it helps companies understand their customers' needs and preferences, identify market trends and opportunities, and make strategic business decisions. It also helps companies minimize risks and maximize profits by providing valuable insights into consumer behavior and market demand.

## How much does marketing research cost?

The cost of marketing research can vary depending on the scope and complexity of the project. Small businesses and startups may be able to conduct their own research with minimal costs, while larger companies may need to invest in professional research firms or use advanced techniques such as focus groups or surveys, which can be more expensive.

## What is the difference between cost and revenue in marketing research?

Cost in marketing research refers to the expenses incurred in conducting research, such as hiring a research firm, purchasing data, or conducting surveys. Revenue, on the other hand, refers to the potential income or profits that can be generated from the results of the research, such as identifying a new target market or launching a successful product.

## How can marketing research help increase revenue?

Marketing research can help increase revenue in several ways. It can provide insights into consumer needs and preferences, allowing companies to develop products and services that better meet market demand. It can also help identify new target markets and opportunities for growth. Additionally, marketing research can help improve marketing strategies and messaging, leading to increased sales and customer retention.

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