Cafeteria Seating Arrangement Problem

Nancy Maple Ray Shannon Christine Bolte Kathy Padgett Jehan Khan

Class: Geometry.

Materials: Measuring devices and included data sheet.

Goals: To give students practice on area, problem-solving, and mathematical modeling write-ups and presentations.

Time Required: Two days. One day will be needed for the students to go to the cafeteria for measurements, and then work together on their calculations. Another day will be needed to collect the write-ups and to hear the presentations.

Background: Prior knowledge of measuring distance, and finding the area of squares, rectangles, and circles.

Setting: The school has decided to replace all of the cafeteria tables and chairs with new ones. Below is a list of the prices and dimensions of the tables and chairs chosen from an office furniture catalogue:

 Furniture Cost Round 60" Diameter Folding Table \$208.60 Square 36" x 36" Folding Table \$96.00 Rectangular 36" x 72" Folding Table \$137.80 Bench Table 54" x 96" \$412.00 Four Chairs 193"w x 302"h x 19f"d \$194.00

Problem: The students are to decide the number of chairs and the number and style of tables that should be purchased in order to maximize seating capacity and minimize cost. The students should also draw a diagram or clearly explain how the tables should be arranged in the cafeteria. Each group should submit a data sheet and a written report, and will present their results to the class.

Evaluation: One grade will be given to the entire group. Point values will be assigned for the data sheet, written report, and presentation. Example: data sheet/report is worth 15 points and presentation is worth 5 points. Note: bonus points may be awarded for the best solution.

Extensions: The problem may be modified by giving the students a budget, rather than having them find the least cost. For a more challenging project, have the students research the price list information. Another alternative would be to require a scaled diagram. Finally, a similar problem would be that the students are to arrange the desks/tables in their own classroom to better suit their needs.

Teacher Notes: This project would work well for groups of three to four students. Be sure the expectations for the written report and presentation are clear to the students before beginning the project. In order to avoid taking the class down to the cafeteria, the teacher may provide the dimensions for the students.

Assumptions - There are many assumptions that could be made about your specific cafeteria. Is it important to have the maximum number of seats or are there social considerations that your school has the luxury of being able to entertain? Do the tables need to be used for other purposes? Is school enrollment expected to be constant?

Report - must include a description of the problem and how it was approached, a list of assumptions, calculations, a concluding statement and/or diagram explaining the arrangement, and the data sheet.

Presentation - should discuss everything in the report.

Cafeteria Seating Arrangement Problem

Sample Report

Project Description: Our school has decided to replace all of the cafeteria tables and chairs with new ones. We were given a list of the prices and dimensions for the chairs and for round, square, rectangular, and bench tables. Our problem was to decide the number of chairs and the number and style of tables that should be purchased in order to maximize seating capacity and minimize cost. Using a tape measure, we found the dimensions of the cafeteria to be 32' x 84'. We then tried four different ways of arranging the tables, and determined the number of people each arrangement would seat. Finally, we calculated the cost per student to predict the most economical table organization.

Assumptions: The following is a list of assumptions we made during the project:

1. The number of chairs around the round tables can be determined by finding the circumference and then allowing two feet per chair.

2. A two foot area is needed around each table to allow for chair movement.

3. We will not put two square tables together to form a 3' x 6' rectangular table, since the cost of a rectangular table is much less than the cost of two square tables.

4. We will put two rectangular tables together and have a person seated at each end.

5. Our school has an enrollment of approximately 900 students.

6. For bench tables we will need to leave at least a two foot walkway between the tables.

7. Each bench table seats eight people.

8. The bench tables do not fold.

9. We do not need to clear the cafeteria for dances.

10. Our lunch period is divided into three periods, each serving 300 people.

Calculations: Since we measured the dimensions of the cafeteria in feet, we first converted the dimensions of the tables to feet and then made the following calculations.

Round  60/12 = 5 (5' diameter)

Square  36/12 = 3 (3' x 3')

Rectangular  36/12 = 3, 72/12 = 6 (3' x 6')

Bench  54/12 = 2.5, 96/12 = 8 (2.5' x 8')

Round Tables: To find out how many people a round table seats, we found the circumference and divided by two, allowing for a distance of two feet per person.

C = (3.14)(5)  16; therefore, seating capacity = 8 people

Since the diameter is five feet and we allowed a two foot clearance around the table for seating, we used nine feet as a diameter in our calculations. To find the number of tables that could fit lengthwise, we divided the length by nine. Similarly, we divided the width by nine.

82/9  9, 32/9  3

This means the cafeteria could be filled with nine rows of three tables. By offsetting three tables, we found that we could fit one more row of tables, for a total of ten rows of three. In other words, the cafeteria could hold thirty round tables. Since each table holds eight people, the seating is 240. Because we assumed that 300 people will be served each lunch period, this arrangement is not acceptable.

Rectangular: Since we have allowed two feet for each person and the length of the table is six feet, three people may sit on each longer side. Also, we assumed that two rectangular tables are pushed together and that one person may sit at each end. This means two rectangular tables pushed together will seat a total of fourteen people. Allowing two feet on either side of the width of the table, we divided the length of the room by seven feet.

Allowing two feet on either side of the length of the table, we divided the width of the room by ten feet. We found that the cafeteria could hold 48 tables. Since each grouping holds fourteen people, the total seating capacity is 336. Each package contains four chairs, so we would need to order 84 packages.

Total Cost of Tables = Number of Tables * Cost of Table = 48(137.80) = \$6,614.40

Total Cost of Chairs = Number of Packages * Cost of Package = 84(194) = \$16,296.00

Total Cost = Total Cost of Tables + Total Cost of Chairs = 6,614.4 + 16,296 = \$22,910

Cost Per Student = Total Cost/Number of Students = 22,910/336 = \$68.19

Square: Allowing two feet for each student, we can seat four people at each table. With two feet added around each table, we divided the length and width of the room by seven. We found that it was possible to fit twelve rows of five tables with one table pushed against the wall in each row. The resulting seating capacity is 216, which is not sufficient according to our assumptions.

Bench: After analyzing several situations, we found that twelve rows of two joined tables could fit perpendicular to ten rows of two joined tables in the cafeteria for a total of forty-four tables. Based on our assumption that eight people may sit at each table, the seating capacity is 352. Since bench tables have built-in seating, no chairs are needed.

Total Cost of Tables = Number of Tables * Cost of Table = 44(412) = \$18,128.00

Cost Per Student = Total Cost of Tables/Number of Students = 18,128/352 = \$51.50

Conclusions: Although there are more possible arrangements and combinations, we have concluded that using bench tables allows for the greatest number of students to dine at one time for the lowest per student cost. The bench tables should be arranged in twelve rows of two joined tables placed perpendicular to ten rows of two joined tables. Because our cafeteria is not used for school dances, we were not required to purchase folding tables.

Cafeteria Seating Arrangement Data Sheet

Sample Solution

 Team Members: Nancy Maple, Jehan Khan, Raymond Shannon

M O *

diameter 60" dimensions 36" x 36" dimensions 36" x 72"

cost \$208.60 cost \$96.00 cost \$137.80

dimensions 54" x 96" dimensions 193"x302"x19f" (including benches)

cost \$412.00 cost \$194.00/package

 kind of table # of tables # of chairs # of people cost per pupil total cost round 30 240 240 \$74.58 \$17,898 rectangular 48 336 336 \$68.19 \$22,910 square 60 216 216 \$75.17 \$16,236 bench 44 0 352 \$51.50 \$18,128

Assumptions:

1. Allow 2 feet per chair around each round table.

2. A 2 foot area is needed around each table to allow for chairs.

3. Rectangular tables are pushed together and 2 people may sit at each end.

4. Two square tables will not be pushed together because of the cost.

5. The school has an enrollment of approximately 900 students.

6. Leave at least a 2 foot walkway between bench tables. Each bench table seats 8 people.

7. The cafeteria does not need to be cleared for dances.

Cafeteria Seating Arrangement Problem

Student Data Sheet

 Team Members:

M O *

diameter _____ dimensions ____x____ dimensions____x____

cost \$______ cost \$______ cost \$______

dimensions ____x____ dimensions ____x____x____

(including benches)

cost \$______ cost \$_________

 kind of table # of tables # of chairs # of people cost per pupil total cost

Please list all of your assumptions on the back of this paper.

Funded in part by the National Science Foundation and Indiana University 1995