Can anyone help me with a lab report for Math Algebra?

The Lemonade Stand

Names: Date:

About this Laboratory

In this laboratory, you will be introduced to a graphing utility. It is important that you become efficient at entering expressions and collecting information from tables and graphs using this Grapher. In order to help you develop your mathematical reasoning skills, the course will emphasize connections among numbers in tables, algebraic analysis, and graphs. In this laboratory, you will learn how to graphically and numerically find and/or approximate the coordinates of points. It is important to familiarize yourself with the techniques of adjusting the viewing window.

The Grapher in this applet requires that * be used for all operations of multiplication. For example, 3(2x + 1) must be typed as 3*(2*x + 1). As you progress, you will learn new technological techniques needed to complete future laboratories. You should begin a reference folder containing your completed labs. Bring the folder with you to the laboratory.

****************************************************************************** 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. • Select Equat. • Enter the right hand side of your equation next to Y1 = . • Select Enter and then OK.

The Lemonade Stand - Pyzdrowski 7/16 -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?)

• 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 set 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 Math Department 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) =

The Lemonade Stand - Pyzdrowski 7/16 -2

Now enter this equation into the Grapher.

• Select Equat. • Enter the right hand side of your equation next to Y2 = . • Select Enter and then OK. • In order to see both graphs and their point of intersection, you should select Out a few

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 unit)

Directions to find the point of intersection: To graphically approximate the coordinates of a point, it may be necessary to adjust the viewing window.

The following is similar to making a box using a graphics calculator. • Move the arrow inside the graph window. • Position the arrow to an area slightly above and to the left of the point. • Press and hold the left mouse button and drag the zoom frame to an area slightly below

and to the right of the point. Release the mouse button. • The graph window should now display the selected region. • Continue to zoom in until an acceptable solution is obtained. In our case, we want the

Xscale: and Yscale: located on the bottom toolbar of the applet to be less than 1.

The Lemonade Stand - Pyzdrowski 7/16 -3

( , )

To read the coordinates of the point: (to the nearest unit) • Point to and Select the point of intersection. • Record the displayed coordinates which appear at the bottom left of the Grapher.

Zooming Back Out:

• Select the Back button to retrace your steps one at a time. • You may also select the Home button on the Grapher to return to the computer’s default

initial setting.

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.

****************************************************************************** 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: • Graph the profit function and sketch all three of your graphs on the same axes. (Identify

each.)

The Lemonade Stand - Pyzdrowski 7/16 -4

• 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?

The Lemonade Stand - Pyzdrowski 7/16 -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 button in the applet to assist you. For example to find R(0), C(0), and P(0): Select Table. Fill in 0, and then select Enter.

x R(x) C(x) P(x)

0 0 25 -25

10

20

30

40

50

60

61

62

The Lemonade Stand - Pyzdrowski 7/16 -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. Be sure to use features of the table, graphs and equations as appropriate for full credit.

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.

The Lemonade Stand - Pyzdrowski 7/16 -7