SUMMER INSTITUTE 2012 BRIDGE & MAZE DESIGN Team Members: (left to right) Ai Choo Ashe Shelley Cady Lisa Boettcher Jim Hamblett Martha Harrison

PROBLEM #1: Algebra-in-Action Bridge Design Construct a bridge using truss design. The span must be greater than or equal to 100 cm.

THE PROCESS: System Analysis Took inventory of all our supplies; listed the dimensions & number of beams and joints, and reviewed directions. Upon the recommendation of our tech teacher, we decided to build a Warren style truss bridge (strength & stability). The art teacher drew 2D and 3D renderings of a Warren bridge to determine if we had enough supplies for our design. She made several drawings before she said, “This is the bridge we can build.” Construction was next with a few adaptations to our original picture. Force was tested with the instruments provided. This testing brought up new questions (See next slide). The word problems and questions came from our experience with the construction and testing of the bridge.

THE PROCESS: System Analysis From the model determine the following: What is the height of the truss? What is the width of the bridge? Find the critical force where bucking occurs and write an inequality Use the Pythagorean theorem to verify the right triangles in the truss design. With a scale of 1 cm. = 10 m, find the actual dimensions for each beam. From the span of the bridge, do the following: Write an equation to determine the number of beams used for the bottom chord Write an equation to determine the number of beams used to find the top chord. This equation must work for any given span. Write an equation to determine the number of joints you will need to construct the bridge. Why is there a discrepancy in the force when a weight sits on the track versus when it is hung from a horizontal beam? Use percent of change to show the result of this experiment mathematically.

THE PROCESS: The Variables Discuss how the pertinent variables that affect the strength of the bridge: The load: the more triangles in the design, the stronger the bridge. Truss height: Will change (a) the hypotenuse Diagonal (a) the hypotenuse: as the diagonal gets longer, the number of joints decreases and therefore, the strength of the bridge which will also affect the maximum load. Cross section- square, round or I-beam will affect the strength of the bridge: I- beam being the strongest. Force (load) to be supported: consider the following.. 1.What kind of traffic will the bridge need to accommodate? Trucks can now carry heavier loads 2.Environmental issues such as: Is this bridge in an earthquake zone? Materials used in construction will affect the strength of the bridge. F max the maximum weight at failure: a safety factor of 2 is usually applied in all constructions.

CALCULATIONS: 1. 4(.17m)(π 2 )(2.3x109N/m 2 )(6.9x10-10m 4 ) / (.24m 3 ) is approximately 770 N Adjusting for the gravity and safety 770/9.8 x ½ Maximum load is approximately 39 kg 2. l 2 + h 2 = a 2 l 2 =a 2 - h 2 h 2 = a 2 -l 2 3.The track becomes another beam and also exerts a force. 4. Proportion: 1 cm = length of the given beam in cm 10m x Actual height is h=170 meters, and a= 240 meters 5. m= 4 (s/L )-3 21 per side 6.2j = m+3 j= 2S/L 12 per side

EXTENSION: Cost and profit of the project. Given values for prices. What is the total cost? Is the Warren bridge the most cost effective for the load it needed to carry? Would another style have been cheaper? Discuss and verify.

REFLECTIONS: Project was valid because: Incorporated all our expertise: tech teacher: prior knowledge of bridge construction, art teacher: graphical visualization of bridge design, math teachers: wrote equations and verified numbers Activity is relevant and applicable to classroom setting where students of varied talents can work together towards a common goal. Activity is relevant for learners who enjoy hands-on manipulations

MAZE PROJECT: System Analysis Problem: Design a maze in a shipping terminal

VARIABLES TA L W CA L c W c CSS L s W s CoS GA GW PW TW TL TH LUA AA TLA RTA CSA RMA N c N s TW TL TH U a

EQUATIONS TA= LxW C A = L c x W c CSS = L S x W S 1.25( L c ) x 1.25(W c ) CoS= n s x CA = n s x L s x W s GA= f (fraction) (TLA) RTA= g(TLA) LUA= a% (TA) AA = b% (LUA) RMA= c% (TA) TLA=d%(TA) CSA= e%(TLA) Unit area = ( 65/61L c + PW) (61/67(W c ) + PW) PA = TA – N s (L s x W s )

INFORMATION: RESOURCES & CONSTRAINTS T otal A rea = 2,000,000 ft 2 TEU = 20 x 8 x 8.5 CA = 160 ft 2 LUA =.06 ( 2,000,000) a=6% 120,000 AA=.03 (TA) b=3% 60,000 RMA= 4% (TA) 80,000 CSA= 3% (TA) 60,000 GA= 2%(TA) 40,000 TLA= 12% (TA) 240,000 RTA= 4%(TA) 80,000 1 Unit = includes the container, slot, and pathway. Size is 944 ft Pathway has to be 35 ft wide to allow for the length of the container and lift truck df= 14 – 11 df = 3

DESIGN A MAZE FOR ROBOTIC LOADING & UNLOADING CONTAINERS Given resources and constraints:  T otal A rea = 2,000,000 ft 2  TEU = 20 x 8 x 8.5  CA = 160 ft 2 ]  LUA =.06 ( 2,000,000) a=6% 120,000  AA=.03 (TA) b=3% 60,000  RMA= 4% (TA) 80,000  CSA= 3% (TA) 60,000  GA= 2%(TA) 40,000  TLA= 12% (TA) 240,000  RTA= 4%(TA) 80,000  Pathway has to be 35 ft wide to allow for the length of the container and lift truck  df= 14 – 11 df = 3

MAZE DESIGN PROBLEM A 2,000,000 ft 2 area is available for the construction of a new maritime shipping terminal. Loading/Unloading area is 6% of Terminal area. Administration area is 50% Repair/Maintenance area is 4% Truck Loading/unloading area is 12% Chassis storage area is ¼ Gate Area is 1/6 Rail Terminal is about 1/3 Design a maze for robotic loading and unloading containers.

PERTINENT VARIABLES The size of the pathway was not given as a fixed value. We had to know the amount of space needed by the robot, forklift, etc. in relation our Unit (container, slot, pathway). This pathway size also affects the number of slots available to store containers. None of this could be calculated until a pathway size was defined. Most of the terminal related to the TA available for building. Decisions had to be made about how the pathways were set up to accommodate loading and unloading the containers. Would the robot need to maneuver in between the containers or would some containers touch the fence on one or more sides, etc. Due to the forklift, the pathway must accommodate the length of the container (20 ft)

WORD PROBLEM: A 2,000,000 ft 2 area is available for the construction of a new maritime shipping terminal. Loading/Unloading area is 6% of Terminal area. Administration area is 50% Repair/Maintenance area is 4% Truck Loading/unloading area is 12% Chassis storage area is ¼ Gate Area is 1/6 Rail Terminal is about 1/3

WORD PROBLEM QUESTIONS 1.What area is available for storage containers? 1,320,000 ft 2 1,320,000/ 944 = 1398 units can fit in the remaining area designated for storage. 2.Change container length to 40ft, how would this affect the number of containers? 3.How many additional units could be stored if the RTA was reduced by 50%? 4.The storage fee for containers is $12.00 per container and a maintenance fee of $25.00 everyday. You begin with 1400 containers on day 1. If you can ship 350 containers per day, what would your total rental fee be for 4 days? Draw a chart for day 1, 2, 3, and 4. 5.Write as equations first. Given total of 1400 containers, solve for the rental fee.

DELIVERABLES #4 Last problem: What is the maximum capacity of storage for you shipping terminal? 66% of TA = 1,320,000 ft 2 space available for storage and pathways CoS = n s x L s x W s PW A = ft 2 CoS = 2078 x (65/61)20 x (71/61)8 CoS = ft 2 Our design allowed every two containers in slots to be placed next to each other without a pathway. This increased our number of containers to Storage Area 1 is 439 ft. x 682 ft. to accommodate 1116 containers. Storage areas B and C measure 200 ft. x 800 ft. each. Each accommodates 481 containers.

REFLECTION We had a better understanding of the system analysis process for this project. The experience of the first one allowed us to work on this project. We were able to apply more creativity to this project such as the 3D representation of our design. It was rewarding to see the numbers jive with our terminal plan. Our Technology teacher explained the design of the maze as the same as the aisles of a grocery store. This visual and use of prior knowledge gave us a better understanding to our maze problem. We applied the math to calculate maximum storage capacity and considered the practical applications of unloading and restocking containers at a terminal. Containers come in and go out constantly in the real-life problem.