The Ice Cream Project
[ Back to Danielle's Math Links Page ]

Show all of your work and fully explain your thinking process for each question. No credit will be awarded for answers with no work shown!

The data below is about ice cream consumption during 10 weeks of spring. The consumption is measured in number of scoops per household per week, the price is measured in dollars per scoop, and the temperature is the weekly average temperature measured in degrees Fahrenheit.

Week	1	2	3	4	5	6	7	8	9	10
Consumption	2.6	2.7	2.9	3.9	3.7	3.9	4.3	4.4	3.4	3.3
Price	1.39	1.33	1.34	1.35	1.41	1.39	1.40	1.36	1.31	1.38
Temperature	24	32	47	41	56	63	68	69	65	61

1. a. Create a scatterplot of temperature (x) versus consumption (y).

b. Does it look to you as if there is a relationship between the temperature and how much ice cream is consumed?

c. What is the equation of the line that passes between the week 1 and 2 points from your part (a) graph?

d. What is the equation of the line that passes between the week 7 and 8 points from your part (a) graph?

e. Compare your answers from parts (c) and (d). What does this say practically about the situation?

f. Connect the points in your graph from part (a) (do this with the connecting points as shown in the table below). Calculate the slopes between each of the points in your graph from part (a) and record the values in a table.

Starting Temp.	Ending Temp.	Slope
24	32
32	41
41	47
47	56
56	61
61	63
63	65
65	68
68	69

g. Determine the temperatures when the consumption showed the greatest increase and decrease.

2. a. Create a scatterplot of price (x) versus consumption (y).

b. Does it look to you as if there is a relationship between the price of the ice cream and how much ice cream is consumed?

c. What is the equation of the line that passes between the week 1 and 2 points from your part (a) graph?

d. Use you answer from part c to estimate the consumption if the price dropped to $1.29. Do you think this is an accurate estimation? Explain.

Scott (a part-time mathematics professor) decides that he wants to open an ice cream stand and he wants to design a large plywood ice cream cone to go on top of his new building. Being a math professor, Scott decides to model the ice cream scoop part of the design (the top part) with the relationship s(x) = and the ice cream cone part of the design (the bottom part) with the relationship c(x) = |x| - 8. x will be measured in feet.

3. a. Graph the ice cream scoop on a graphing calculator and submit your best reproduction of the graph with your project.

b. Graph the ice cream cone on a graphing calculator and submit your best reproduction of the graph with your project.

c. Is the ice cream scoop relationship s(x) = a function? What is the domain of s(x)?

d. Is the ice cream cone relationship c(x) = |x| - 8 a function? What is the domain of c(x)?

e. Graph the scoop and the cone together on the same set of axes (make this a very large graph as you will be adding to it).

f. Scott wants to put crossed support beams behind the plywood ice cream cone at the x = -3 and x = 3 lines. In order to do this, he needs to know s(x) and c(x) at those points. What are s(-3), s(3), c(-3) and c(3) (be accurate to at least 4 decimal places)?

g. To build his crossed support beams, Scott needs to know how long to make his beams. What is the distance from the point (-3, s(-3)) across to the point (3, c(3))? What about from (-3, c(-3)) to (3, s(3))? Be accurate to at least 2 decimal places.

h. Show what his crossed support beams will look like on your graph from part (e). Label all four points mentioned in part (g).

i. What are the x- and y- intercepts of s(x) and c(x)? Include those on your graph from part (e).

j. At what values of x will the scoop be 6 feet high? Be accurate to 2 decimal places.

k. Design a grid pattern of your choice with at least 8 lines for the cone (horizontal, vertical and slanted lines are okay). What are the equations of the grid lines? How long are the grid lines? Sketch the grid pattern on a graph of the cone including all points of intersection with c(x).

Now, Scott wants to be competitive with his pricing, so he wants to use mathematical modeling to figure out his costs, revenues and profits.

4. Scott bought the ice cream stand building for $2,000.00 and he will spend an extra $200 on the plywood ice cream cone design from problem 3. The monthly rent for the space his building is on is $150.

a. Write a linear equation giving the total cost C of operating the ice cream stand for n months.

b. Assuming that Scott can bring in $350 per month, write the equation for the revenue R derived from n months of operation.

c. What is the formula for Scott’s profit from n months of operation?

d. At what point does Scott break even?

Last Modified on Monday, January 26, 2004.