Market Research
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
355
320
265
235
180
125
Price, p
$1.50
$2.00
$2.50
$3.00
$3.50
$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
5 cups flour
3 Tbs. sugar
2 tsp. salt
¼ tsp. baking soda
1 package dry
yeast
1 cup buttermilk
1/3 cup milk
1 egg
packaging
Total
price/package
size
$3.86 / 19 cups
$4.98 / 378 Tbs.
$0.52 / 122 ¾ tsp.
$0.60 / 100 ¾ tsp.
$0.66 / package
price / single
loaf
$0.508
$0.020
$0.004
$0.001
$0.330
$1.17 / 4 cups
$2.38 / gallon
$2.35 / dozen
$0.146
$0.025
$0.098
$0.050
$1.182
Introduction:
Supply and demand is one of the most fundamental concepts of economics and it
is the backbone of every business. It is very essential to know how much quantity of a
product is desired by a buyer, so that there is a balance between supply and demand and
the business gains a healthy profit. In this project of homemade bread business to get a
handsome profit, we need to decide exactly how much is the demand of the bread so that
we can produce that much loaves. According to the price of ingredients and adding labor
cost, profit etc. to it for deciding the cost per loaves. The cost function is calculated and
the revenue function is to be formulated to make profit. This project is all about studying
and analyzing the law of demand, law of supply, price and demand relationship and
profits.
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?
Demand Function
$4,50
$4,00
$3,50
Price
$3,00
$2,50
$2,00
Price, p
$1,50
$1,00
$0,50
$0,00
0
50
100
150
200
250
Loaves Sold, x
300
350
400
Demand Function
y = -0,0109x + 5,4287
R² = 0,9929
$4,50
$4,00
$3,50
Price
$3,00
$2,50
$2,00
Price, p
$1,50
$1,00
$0,50
$0,00
0
50
100
150
200
250
300
350
400
Loaves Sold, x
Yes, the relationship appears to be linear with inverse relationship. The trend line is a
straight line with negative slope. The line of best fit is shown in the scatter chart with slope,
m =-00109 and y-intercept=5.4287. The demand equation is, y=-0.0109x+5.4287. The
correlation coefficient is, r=-0.99644, which shows that there is strong negative 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.)
Answer:
R(x) =p*x;
𝑥 = number of loaves and 𝑥 ≤ 400
𝑅(𝑥) = 𝑥 ∗ (−0.0109𝑥 + 5.4287) = −0.0109 𝑥 2 + 5.4287𝑥
The revenue function is; R(x) =-0.0109*x2 +5.4287 x
Revenue
$800,00
$700,00
Revenue
$600,00
$500,00
$400,00
$300,00
$200,00
$100,00
$0
50
100
150
200
250
300
350
400
450
Number of loaves sold
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?
Answer:
The oven cost is the fixed cost which can be calculated weekly. The $1.182 is the cost of
the ingredients for a single loaf of bread which is the variable cost.
Weekly Fixed Cost =
203.40
× 7 = $47.46𝐶(𝑥) = 𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡 + 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐶𝑜𝑠𝑡𝐶(𝑥)
30
= 47.46 + 1.182 𝑥,
𝑥 ≤ 400
The domain of the cost function is: 𝑥 ∈ [0, 400]
Cost Function
$600,00
$500,00
Cost
$400,00
$300,00
$200,00
$100,00
$0
50
100
150
200
250
300
350
400
450
Number of Loaves
4. Profit Function. Find P(x), the weekly profit for producing and selling x loaves of bread.
(Hint: profit = revenue – cost.)
Answer:
Profit = Revenue –Cost
𝑃(𝑥) = 𝑅(𝑥) − 𝐶(𝑥) = −0.0109 𝑥 2 + 5.4287𝑥 − (47.46 + 1.182𝑥)
= −0.0109 𝑥 2 + 4.2467𝑥 − 47.46
The profit function is, P(x) =-0.0109x2 +4.2467x-47.46
Profit Function
400
Profit ($)
300
200
100
0
-100
-200
0
50
100
150
200
250
Number of loaves
300
350
400
450
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?
Answer:
The revenue function is; R(x) =-0.0109*x2 +5.4287 x
The function is a parabola with concave down, so the maximum point is the vertex.
The vertex can be found by;
5.4287
𝑥=−
= 𝟐𝟒𝟗𝑅(249) = $676.06
2 ∗ (−0.0109)
The number of loaves to maximize the revenue =249
The Maximum Revenue = $676.06
The Price function is; 𝑝 = −0.0109𝑥 + 5.4287
To maximize the revenue,
𝑝(249) = −0.0109 ∗ (249) + 5.4287 = $ 2.7146 ≈ $𝟐. 𝟕𝟐
The price charged should be $2.72 per loaves to maximize the 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?
Answer:
The profit function is; P(x) =-0.0109x2 +4.2467x-47.46
Again, the function is a parabola with concave down, so the maximum point is the vertex.
The vertex can be found by;
𝑥=−
4.2467
= 194.80 ≈ 𝟏𝟗𝟓𝑃(195)
2 ∗ (−0.0109)
= −0.0109 ∗ (195)2 + 4.2467 ∗ (195) − 47.46 = $ 𝟑𝟔𝟔. 𝟏𝟕
The Price function is; 𝑝 = −0.0109𝑥 + 5.4287
𝑝(195) = −0.0109(195) + 5.4287 = 3.3032 ≈ $𝟑. 𝟑𝟎
The number of loaves that should be produced is 195 for maximize the profit.
The Maximum profit = $366.17
The price per loaves to maximize the profit =$3.30
7. Conclusion. How many loaves of bread will you produce each week and how much will
you charge for each loaf? Why?
Conclusion:
From the demand chart, we can observe that as the price increases that loaves sold
decreases and vice-versa. We need a healthy profit for the business and to pay the loan.
After completing all the calculation we have found to maximize profit there should be
195 loaves of bread produced each week to get a maximum profit of $366.17 and we
must charge $3.30 per loaves. We need to maximize our profit not revenue that is why we
get the above numbers which is important to our business to pay the monthly payment for
the oven and get a good profit.