The Lemonade Stand
Names:
Date:
About this Laboratory
In the following laboratory you will learn how to solve problems graphically. Then you will
compare the graphic solutions to a table of values and also to an algebraic model. You will use
communication skills and the Rule of Three (NAG). By making connections among Numbers in
tables, Algebraic analysis, and Graphs you are using a type of mathematical reasoning. By
working independently with your partner you are discovering these connections and thereby
constructing your own knowledge about mathematics. A teacher who facilitates students’
construction of their own knowledge is sometimes known as a constructivist.
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You are going to sell lemonade at the school fair to earn money for the Math Club.
The selling committee decides to sell the lemonade for 50 cents a cup. Express your monetary
units as dollars.
•
In the space provided, write an equation to express the amount of money you can earn as
it relates to (depends upon) the number of cups, x, that you sell. We call this amount
revenue.
•
Let the amount of money earned be R(x). Because the revenue depends on the number of
cups sold, R(x) is sometimes called the dependent value.
•
R(x) =
Now enter this equation into the Grapher http://www.meta-calculator.com/online/?panel101-equations
•
•
•
Select the Equations tab.
Enter the equation in the blank.
Hit Enter.
WV Algebra I Project: The Lemonade Stand - Pyzdrowski 2/07 -1
Sketch a picture of the graph you see in the plotting window.
•
What does the x variable represent in this example?
(In other words, how would you label the x-axis or the independent values?)
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What does the R(x) value represent in this example?
(In other words, how would you label the y-axis or dependent values?)
•
What x values( what types of numbers) make sense for the example we are using?
The x values that make sense for the problem make up the Restricted DOMAIN.
(To find the restricted domain, it may help to use indirect reasoning. Think -- “Are there any
values of x that do not make sense for the problem?”)
In order to sell the lemonade it was necessary to pay $25.00 to the fair committee to rent the
booth. The lemonade costs $3.75 for a container that makes 50 cups. Cups cost $2.00 for a
package of 100. The students of the Math Club will provide the water, ice, and all other
materials needed to make and serve the lemonade. In the space provided, write an equation for
the cost as a function of the number of cups of lemonade sold. Include in your equation the fixed
cost for renting the booth. (The Prom committee will buy any left over cups and lemonade from
the Math club for the same price that was paid; therefore, you may ignore any problem of
overbuying.)
•
C(x) =
WV Algebra I Project: The Lemonade Stand - Pyzdrowski 2/07 -2
Now enter this equation into the Grapher.
•
•
•
•
Select the Equations tab.
Enter the equation in the blank.
Hit Enter.
In order to see both graphs you may need to click Zoom Out several times.
Make a combined sketch of both graphs, cost and revenue. Identify each on the sketch.
Use the following directions to determine at what point the two lines intersect? (to the nearest
dollar)
Directions to find the point of intersection:
•
•
Select the Intersection tab.
Select the two equations and click Plot Points of Intersection
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Record the displayed coordinates which appear on the graph.
(
,
)
WV Algebra I Project: The Lemonade Stand - Pyzdrowski 2/07 -3
Use Algebra to solve for the point of intersection in the space provided:
Describe the real life meaning of the point of intersection.
(What is so important about these coordinates in terms of cups sold and cost to the club?) You
should use the table that is to follow in order to confirm your answer.
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Profit is found by subtracting total Cost from Revenue. That is: Profit = Revenue - Cost.
•
Write an equation for Profit in this example using x as the independent variable.
P(x) =
Use earlier directions to do the following:
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Graph the profit function and sketch all three of your graphs on the same axes. (Identify
each.)
WV Algebra I Project: The Lemonade Stand - Pyzdrowski 2/07 -4
•
•
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On your sketch, show all three x-intercepts and the point of intersection for the revenue
and cost functions.
Give the coordinates of
a) the point of intersection of the cost and revenue functions,
b) the y-intercept of the cost function and
c) the x-intercept of the profit function.
What is the x-intercept of the profit graph? ( to within 1) (
,
)
What is the meaning of the x-intercept of the profit function in this problem?
WV Algebra I Project: The Lemonade Stand - Pyzdrowski 2/07 -5
Now set your expression for P(x) equal to zero and solve for x algebraically (by hand).
Fill in the table. You may use the Table tab in the applet to assist you. For example to find R(0):
Select Table. Choose the R(x) function and click Create Table. Then change Start x at: 0 and
Interval of x: 10.
x
R(x)
C(x)
P(x)
0
0
25
-25
10
20
30
40
50
60
61
62
WV Algebra I Project: The Lemonade Stand - Pyzdrowski 2/07 -6
Do Later:
1) Write a few concluding remarks about how the 3 graphs, the numbers in the table, and
algebraic analysis about cost, revenue and profit are related. That is, explain how each of the
three representations ( tables, graphs and equations) can lead you to the same approximate
solution for the break even point.
2) If your club wanted to make a profit of $500, how many cups of lemonade would need to be
sold? Give the solution and explain how you arrived at your answer.
WV Algebra I Project: The Lemonade Stand - Pyzdrowski 2/07 -7