Mathematical Modeling in Engineering Design Projects

Mathematical Modeling in Engineering Design Projects: Insights from an Undergraduate Capstone Design Project and a Year-Long Graduate Course Monica E. Cardella CASEE Postdoctoral Engineering Education Researcher Center for Design Research, Stanford University While engineering students are required to complete a number of mathematics courses, some engineering students and practitioners believe that they do not use the mathematics that they learned from their courses in engineering projects. This study investigates engineering students’ use of mathematics through observations of two teams of students working on extensive design projects. The cases studies presented in this paper provide insights into situations when engineering students engage in modeling behavior and also explore ambiguity and precision in engineering design. These insights can inform engineering education as we help engineering students become more aware of the ways that mathematics is used in engineering. Additionally, understanding the ways that mathematics and mathematical thinking is used in professional applications can help us motivate and contextualize mathematics instruction as well as determine what should be taught to students in both college and pre-college settings. Introduction: The study presented in this paper was originally motivated by the disparate perceptions of mathematics that are evident in engineering education. While some engineering students are able to recognize the value of the mathematics that they have studied, especially as it relates to their professional practice (Graves, 2005) and its utility as a “tool” (Satwicz, 1994), other students question the relevance of the courses they have completed (Cardella, 2006). Within professional practice, some engineers also question the usefulness of the mathematics they have studied. As Pearson (1991) reflected on the conversations he had engaged in with thousands of engineers, he estimated that only 30% actually used the calculus and differential equations that they studied in college during the course of their professional projects. However, within engineering education, mathematical and analytical competence is recognized as being a fundamental skill that all engineering students need (ABET, 2003; NRC, 2006). Because of these disparate perceptions, we might wonder why mathematics is important for engineering, and how it is that engineers actually use mathematics. However, the study is also motivated by the opportunity to further understand how mathematics is used in authentic situations (Hutchins, 1996). Understanding mathematics use in everyday life provides opportunities to motivate mathematical learning throughout the learning continuum (i.e. from preschool to university), as well as to better understand the types of mathematical competencies and mathematical thinking that are important for success in every day life as well as specific workplaces. With this in mind, this paper follows in an established tradition of studying mathematics in workplaces (Hoyles & Noss, 2007; Hall, 1999; Stevens & Hall, 1998). One study of mathematics in the workplace that is particularly relevant is Gainsburg’s study of the mathematical practices of structural engineers (2003). Gainsburg observed structural engineers with varying levels of experience at two engineering firms as the engineers when about their usual work. She found that structural engineers live and think in a world of quantities, units, procedures, and concepts (Gainsburg, 2003). She also found that mathematical modeling was central to and ubiquitous in the engineers’ work (Gainsburg, 2006)—the engineers in her study used, adapted, and created models of various representation

forms and degrees of abstraction (2006). Gainsburg describes the practicing engineers’ on-thejob knowledge acquisition (2003), and suggests that this constructivist form of learning and the types of models the engineers worked with are not well-reflected in the modeling tasks typically prescribed for the K-12 classroom (2006). This study builds on Gainsburg work through similar unobtrusive observation of engineers at work in their normal practices—in this case engineering students working with industry partners on 5 to 9 month-long authentic design projects. In addition, this work builds on other studies of engineering students’ modeling behavior and students’ acquisition of modeling skills (Diefes-Dux et al., 2004; Moore & Diefes-Dux, 2004). Methodology: The data and analysis presented in this paper build on a previous study of engineering students’ uses of mathematical thinking in engineering design. The data presented in this paper on the undergraduates’ modeling activities was originally collected as part of a larger investigation of engineering students’ uses of mathematical thinking in the context of capstone (or senior) design projects. In the previous study, the data was analyzed along five aspects of mathematical thinking derived from Schoenfeld’s description of mathematical thinking (1992): knowledge base, problem solving strategies, use of resources, beliefs and affects and mathematical practices. These five aspects of mathematical thinking provided an opportunity to illuminate engineering students’ mathematical knowledge bases in addition to the problem solving strategies they learned from their mathematics courses, the resources that they learned to use in their mathematics courses, the ways that engineering students learned to monitor their use of mathematical resources (i.e. metacognitive processes such as planning, monitoring and reflecting), beliefs and affects engineering students have developed about mathematics and the types of mathematical practices that students engage in (e.g. using estimation, being precise, using numbers to justify a design decision). These findings are discussed in more detail elsewhere (Cardella, 2006; Cardella & Atman, 2004). This paper focuses on one specific mathematical thinking “theme” that emerged through grounded theory methodology (Strauss & Corbin, 1994; Coffey & Atkinson, 1996; Dick 2004) from the previous analyses: mathematical modeling. Two groups of engineering students participated in this study: one team of five Industrial Engineering undergraduates participated during their final year of college, and one team of four Mechanical Engineering students who were pursuing their Masters degrees. Each of these students agreed to be observed during their team meetings over the course of approximately five months. The students also shared copies of the documents that they created for their project— both interim design documents (such as rough sketches) and copies of the final reports. Eight of the students also agreed to be interviewed about their educational backgrounds (particularly their mathematical background) and their own estimation of how they used mathematics in engineering design projects. The data presented in this paper provide snapshots of two of the teams’ meetings—one of the undergraduate team’s meetings and one of the graduate students’ team meetings. These two case studies (Yin, 2003) provide a rich description of two scenarios when engineering students engaged in mathematical modeling as part of an extended design project. In particular, we are able to learn about the different types of models that engineering students create and use as well as the ways that engineering students grapple with ambiguity, uncertainty, and the need for precise information. Industrial Engineering Undergraduates: A team of five Industrial Engineering undergraduates worked together to evaluate and change part of a supply chain system. Like their classmates, they had been paired up with an industry partner who had assigned the team an authentic

problem that they would work on over the course of their five-month capstone design project (see Figure 1). During the course of the five months, the team met regularly with their industry partners—engineers representing the partner—in addition to their weekly meetings with their capstone design course instructor. In order to provide their industry partner with a final recommendation on whether or not they should establish a satellite center on Bellissima Island, the team determined that they needed to do a detailed cost analysis of each option (keeping the system as it was, changing the delivery routes, or adding a satellite distribution center). The team has three additional options to consider; there are three standard models for how the packages would be sorted into individual deliver routes that they might use for the satellite center. Figure 1: Problem Statement for the Industrial Engineering Students Team Currently, six to eleven package cars travel daily from the Supply Chain Solutions (SCS) hub to and from Bellissima Island. During a typical nine hour shift, each of the package car drivers will spend an average of 60 minutes commuting to Bellissima Island. An additional 50-60 minutes will be spent returning to the hub from Bellissima Island. Upon arriving onto Bellissima Island, each package car driver will then travel to their designated loop where they will complete their necessary deliveries and package pick-ups for the day. One of the big concerns SCS has with the Bellissima Island routing process is idle time. A large amount of idle-time is incurred due to the commute to the ferry dock, the ferry queue, the 35 minute trip across the Sound, and the queue to get off of the ferry. Idle-time is defined as the time when a driver is not doing anything that is value adding to SCS. Due to the large amount of idle-time, SCS deems the driver utilization rates for the Bellissima Island delivery loops as unsatisfactory. Ideally, package car drivers should deliver high volume over the minimum number of miles necessary to complete their tasks. In the case of the Bellissima Island routing process, too much of a driver’s day is spent on commuting rather than delivering and picking-up packages. One way to achieve higher utilization rates is to deliver a high volume of packages using smaller distances. The reason for a volume over miles ratio is that it is inefficient and expensive to pay drivers to be idle. Although the underlying goal of this project is to reduce the cost of the Bellissima Island delivery process, it is understood that in order to achieve this, the volume-miles ratio must be minimized. During meeting 141 three of the five undergraduates have gathered to work on their project. One of the first tasks that they attend to is developing a plan for the paper that is due at the end of the project. The end is quickly approaching—they now have two weeks left to finish the project before presenting their solution to a panel of judges and providing their industry partner with a recommendation. After developing a tertiary plan, they turn to the issue of utilities. In analyzing the cost of setting up a satellite center on Bellissima Island, one of the many costs they need to include is utilities. Diego reviews the utilities they need to consider: garbage rates (and the initial cost of purchasing garbage cans), electricity (which means they also need to determine how much power is required for the satellite center’s daily operations), “honey bucket” prices (and the cleaning rate for the honey buckets, where the “math is: 5 guys working eight hour days”)                                                              1

 Meeting 14 was the 14th meeting that I observed. 

and taxes. Diego also notes that if they were to rent a facility, many of the utilities would be covered under a single service fee. They consider the potential for recycling—which could reduce the cost of garbage but would increase the initial cost associated with purchasing garbage cans. They also talk about the sewage services: if the satellite center is position near a gas station, perhaps they do not need to pay for portable toilet services. Following the discussion of the utility costs, the team returns to an issue they have discussed in earlier meetings: the population growth rate. As the team evaluates the options for delivering packages on Bellissima Island, they have realized that they must consider not only the current patterns in the number of packages delivered at various times of the year, but also the future quantities of packages that will be delivered on the island. In conversations with one of their industry partners, they learned that the growth pattern for the package volume (the number of packages that are delivered) closely mirrors the growth pattern for the population of the island. After revisiting the growth rate the team moves on to consider one of the project’s major constraints. In addition to considering the costs of utilities and land, and considering quantities of packages involved in the system, the team must also consider human factors. In particular, the team must ensure that the delivery drivers are not working unrealistic hours. Legally, no delivery driver can work more than 12 hours in a single day. The team has considered having one driver transport the packages from the main distribution center hub to the satellite center at the beginning of the day and then have the drive return to the hub with all of the packages that have been picked up in the course of the day. In addition to considering the number of hours a driver can legally work, the team is also considering other time constraints: the time in the morning when the packages will be ready to leave the hub for transit to the island, the time in the evening when the packages must be back at the hub if they are going to be able to be shipped out that night, and the guaranteed deadlines the company has established for its customers for package delivery and pick-up. To determine if they are meeting the time constraints, the team needs to understand how much time is required for each task that is part of the delivery and pick-up system: the amount of time required to transport packages from the hub to the island, the time required to sort the packages brought over in one large transit truck into five, six or seven delivery routes, the amount of time spent delivering all of the packages (and picking up new packages), the amount of time spent loading all of the picked-up packages onto the truck that will return to the hub, and finally the time required to return to the hub from the island. The students have received information from their industry partner that will help them answer some of these questions: they have been given an Excel spreadsheet that contains several months of data for the number of packages delivered each day and the amount of time spent on each individual delivery. With this data, and some additional information from the industry partner, they have been able to create simulations of the sorting process to determine a range of values for the amount of time spent sorting the packages into the different delivery routes. They have also analyzed the data to determine the amount of time spent delivering packages (they have determined the average time spent delivering packages, as well as the range of amounts of time drivers have spent). At this time, during meeting 14, the team is checking their previous work. Is it possible that we overestimated the time needed to sort the packages into the different delivery routes? They are finding that the constraint of the 12-hour day is not easy to meet. They consider hiring additional part-time drivers to help sort the packages. What happens if we run out of time? Will the delivery driver be stuck on the ferry? John checks the ferry schedule for the fare information as well as

the schedule of times; one concern is the frequency of the ferry trips, and the implications of the driver missing the appropriate ferry. They also consider all of the distances that will be driven. Are these distances rectilinear or Euclidean? We need accurate distances. Diego continues estimating the distances while Mei and John try to envision the daily schedule. John looks for information off the internet while Mei begins to write out the daily schedule for the system of all of the transit and delivery cars. At 1:00am John decides that he is no longer able to contribute; Mei continues work on the schedule on a whiteboard while John goes home. Mei considers the drivers’ lunch breaks, and revises the schedule as she writes. She considers the drivers’ current schedules, and the fact that the time that they currently spend on the ferry they can spend delivering packages in the satellite center scenario. Because the 12-hour limit is not negotiable, she takes a conservative approach and plans the day so that the driver does not work more than 11 hours each day. When she reaches a satisfactory schedule, she copies down her work and explains that she based much of her work on the current delivery schedule and specific time constraints, such as the ferry schedule and the hub’s typical schedule (the time of day when packages are ready to leave the hub and the deadline for when they must return to the hub). Beyond that, she looked at “one section at a time” to make sure that all of the constraints were met. She and Diego are content with their progress and leave the undergraduate meeting space to go home. Mechanical Engineering Graduate Students: A team of four Mechanical Engineering graduate students worked to design a portable dental unit that dentists could take to use in remote rural locations, or use in a small private practice. Like the undergraduates, the team worked with an industry partner—a company that manufactures compression and vacuum systems for dentists. The main criteria that the team considered in their design decisions were weight of the system, cost of the system, and performance of the system. Specifically, the team worked to ensure that the compressed air and vacuum delivery system was able to provide a certain air flow in order to power the dental tools: the drill, suction and spray. To ensure that their system met the air flow constraint, the team created two models of the system: physical, “experimental” models and mathematical “simulation” models. With the experimental models, the team connected motors, compressors, vacuums and dental tools to test the systems’ ability to meet the constraints. Using Engineering Equation Solver, the team created mathematical models of the same systems to simulate the air flow through system. During the February 20 meeting, the team is looking at the “theoretical” model (the simulation) and interpreting “what the numbers mean.”They are investigating the ways that different piston sizes and layouts (in the motor powering the compressor) effect the rate of air flow. They realize that at any pressure, q[in]=q[out]: the air flow going in to the system should equal the air flow exiting the system. The team discovers two potential problems: 1) air flow in does not always equal air flow out and 2) the air flow they measured in the experimental models did not always match the airflow predicted by the simulation. They run a simulation and are trying to understand the correlation between increases in pressure and decreases in air flow (scfm). Scfm is the air flow at standard conditions, and should be consistent—q[in] should equal q[out]. One of the course instructors, who acts as a technical advisor for each of the eleven teams in the class, explains one potential cause for the discrepancy: a gap in the chamber by the piston which leads to a growing air pocket as pressure increases, causing the air intake to decrease. When they finish compressing air, the air that is trapped in this gap expands and also sucks more air in.

As the team talks with the instructor about the discrepancy between the experimental results and the simulation results, the instructor iteratively checks the experimental set-up for problems— tubes that are too long, mistakes in the order or direction of the equipment and mistakes in how air flow is measured—and checks for errors in the mathematical model. He checks the physical devices, questions the team on the mathematics and returns to the physical devices. He is called away to work with another team, and so our team continues the process of checking both the experimental set-up and the mathematical model for potential mistakes. The team decides to collect more experimental data. Two of the teammates set up the equipment—they build a base to hold one measurement device, prepare the press and set up their laptops with LabView. As they begin collecting data, they have a target pressure of 100psi and a maximum flow rate of 1.69scfm. They start the drill press, look at the data that is being collected via LabView in an Excel spreadsheet and then stop the drill press when they have reached the target pressure. Initially they collect data at 1600 rpm, and decide to collect it at a few other speeds too—they collect some unplanned data because everything is already set up and they want to make sure they have everything they need. They collect data, reflect on the data—the starting values, the maximum values, etc.—then collect more data. They revisit their constraints—the hand piece a dentist uses requires 1.5 scfm of air flow, and they are planning to supply a total of 3 scfm. They consider that the highest air flow needed would be a combination of a hand piece—which they now quote at 1.4scfm—and a vacuum, which needs 1.97 scfm. They are not confident in their recollection of this constraint, and remind themselves that they need to look at the matrix of dental unit usage that the industry partner gave to them—they think there may be a maximum of three devices used at a time, where one is the vacuum (“always have the vacuum on”) and the dental assistant may be running spray in addition to the dentist’s use of another hand piece. After considering these constraints, they turn to modeling tasks—one team member asks if they have modeled their newest compressor yet. They decide they need to open the compressor up to measure different parts, and they run the system with LabView to measure the air flow and pressure. One team member looks at the specifications for the compressor on the manufacturer’s website, and realizes that the main air flow rate that is advertised is unrealistic; it is only achievable under specific, rarely realized circumstances. They redirect their focus to the simulation they are creating using EES (Engineering Equation Solver). They check their units of measurement, and realize that they have not been consistent. They also check the mechanical deficiency that they have been using. Their instructor had previously shared an academic paper with them that served as a valuable reference for determining the appropriate mechanical deficiency. They work through by hand some of the calculations that relate air flow, mechanical deficiency and pressure, and debate whether they should only include the final value for the air flow output in the code, or if they should also include the pressure, mechanical deficiency and the air flow entering the system, since these three values would be captured in the value for air flow out. So, rather than having the code calculate that q[in] will be 7.5 scfm every time they run the script, they consider setting q[in]=7.5 scfm in the code because this is “what the code needs to know.” “This dictates the geometry of our pistons.” This discussion around the calculation of air flow brings them back to their initial conversation about air flow: q[in]=q[out]. At this point, they realize that although their q[in] will always be the

same, their q[out] will “always be changing” (though it will be consistent for a particular pressure). The deficiencies—which will cause q[in] to not equal q[out], will “correlate to the geometry of the pistons.” The final values that need to be inputted in to the mathematical model for the simulation are the dimensions of this particular compressor. They begin by estimating the stroke length, piston diameter and speed (number of rotations per minute). They turn to the internet to see if they can confirm the estimates, but in the end determine that they need to open up the housing to measure the parts themselves. The stroke length is 1.2 inches (although when they write this number down, they change it to 6/5”) and the piston diameter is 1.75 inches. As they continue to move in between their theoretical model and their measured experimental results, they compare the stroke length from the simulation – 1.8 inches—to the “actual” stroke length: 1.2 inches. They note that they were off by half an inch, and decide that designing it with an extra half an inch is reasonable. However, they still double check the stroke length. They realize that the value in the simulation is not 1.8 inches but is actually 1.28 inches. In the end, they decide that the simulation is correct—or at least close enough. They turn to other tasks, such as the recent email they have received from their Industry Partner and further analysis of the data they collected near the beginning of this design session. While his teammates look at a graph they have created, Daniel revisits the results from the experimental and simulation models (he is still concerned about the discrepancy, even though he had earlier decided it was close enough) and realizes that he had forgotten something. He fixes the error, as the team concludes the design session. Discussion: Both teams’ projects involved multi-layered models. The team of industrial engineering undergraduates ultimately modeled the cost of each of the alternative solutions that they considered for the package distribution problem. To model the cost of the proposed satellite center solution, however, they needed to complete three other modeling tasks. One of the factors associated with the cost of the satellite package distribution center was the cost of drivers’ salaries and wages. To include this cost in their analysis, and to ensure that the satellite center was a feasible alternative, the team created a model of the daily schedule for all aspects of this solution—the times spent on each subtask that needed to be completed for all packages to be delivered (and picked up) on the Island. To model the daily schedule, the team needed to know how much time would be spent each day on the sorting process (sorting the packages brought to the island by the transit car into the different delivery routes) and how much time would be spent during the delivery portion of the day. Each of these times—the time for sorting and the time for delivery—were determined by creating mathematical models of these processes using data and information the team acquired from their industry partner. The team of mechanical engineering graduate students attended to the task of modeling air flow and pressure in the design of an air compression and vacuum delivery system that would be used to operate dental tools. To create this model, the graduate students, like the undergraduates, attended to several sub-tasks that involved modeling. In particular, the team’s system consisted of a combination of a motor, compressor, vacuum and the dental tools. The team considered multiple options for most of these components and deemed that it was important to model each individual component in addition to the system (for example, the team created mathematical models for multiple compressors). The mechanical engineering graduate students also created

multiple types of models for the entire air compression and vacuum delivery system—they created “experimental,” physical models as well as “theoretical,” simulation models. One notable difference between the two projects was the tangible aspect of each project. The mechanical engineering students were able to construct physical models of the system to use to further understand the system as well as to check the accuracy of the mathematical models. The industrial engineering students were not able to build a satellite center or otherwise physically interact with prototypes of their designs. They were, however, able to observe the current package distribution system as it operated and took advantage of the opportunity to follow the delivery drivers through a day of work. Additionally, when the team approached the task of modeling the daily schedule for the proposed satellite center solution, they took a nonmathematical approach by writing and re-writing the schedule and using an intuitive sense of checking to see if constraints were violated, rather than creating a linear program to solve the scheduling problem. One of the challenges that the undergraduate Industrial Engineering students commented on during the course of their project was the need for “precise information” in order to have a precise final cost for each alternative solution, which was frustrated by ambiguity and missing information. During meeting 14, some of the information that was missing was the cost of specific utilities as well as information about which utilities were actually necessary. In an interview after the project was over, one of the students commented that perhaps he and his teammates should have used a variable as a placeholder for the missing information, rather than allowing the missing information to impede their progress on the project. At other times, the team overcame the problem of missing information by using estimates. However, they later became frustrated that the estimates were not precise. As the undergraduates struggled with ambiguity, uncertainty and estimation, it is interesting to consider what they may have believed “precise” meant. From their behavior, the Industrial Engineering students presumably envisioned an exact dollar amount that would not change. However, from a mathematical perspective, representing the costs of each utility with a variable (as the utility costs may vary over time and as the company grows) as well as the other factors such as employee wages, may be a more precise solution. The graduate students’ activity provides another interesting perspective on precision. As the mechanical engineering graduate students verified that the proposed compressed air and vacuum delivery system would provide enough air for the dental tools to function, they considered many sources of information. They evaluated the system by collecting “experimental” data and also evaluated the system through a mathematical model and “simulation” data. Later, when the two did not match, they iteratively checked each model—with an underlying question of which model is more accurate, more precise? Likewise, in creating the mathematical model, they gathered information from multiple sources: estimates of lengths and diameters, manufacturer specifications and their own measurements of the stroke length and piston diameter. Ultimately, they too needed to determine which source was most accurate, most precise. It is interesting to note too that the graduate students tended to gather information from multiple sources—perhaps suggesting that no single source was accurate or precise enough to rely on. Conclusion: Like Gainsburg’s study of structural engineers, this study provides evidence that mathematical modeling is central to engineering practice and a valuable tool for engineers. While the students may not have used every particular aspect of the mathematical knowledge base they

had acquired throughout their undergraduate and pre-college studies, creating mathematical models was integral to their ability to complete their project. At times the students explored alternatives to mathematical models—at times to understand the project from a “hands-on” perspective, at times to simply get a second perspective on the problem, and perhaps at times the non-mathematical approach was more efficient. While one of the undergraduates reflected on ways that his team might have used mathematical modeling more in their project, it seems that each team had some understanding of how mathematical modeling could help them complete their project (and also understood some alternatives to mathematical modeling). The cases in this paper represent examples of how engineers use mathematics, which can provide context and motivation for mathematics learners: undergraduate engineering students, undergraduate mathematics students and pre-college students. The cases also provide insights into further opportunities in undergraduate engineering education. Both teams also grappled with the ambiguity and uncertainty inherent in their projects, and struggled to be accurate and precise. The two cases presented in this paper provide insights into students’ perceptions of “precision,” and suggest that some undergraduate engineering students can become frustrated by the ambiguity and uncertainty that are normal for authentic engineering tasks. The data in this paper provides further support for the notion of introducing students to complex, authentic tasks that will give students practice with responding to uncertainty and ambiguity. Acknowledgement: This research was made possible in part by National Science Foundation grant SBE-0354453. This work was supported by the National Academy of Engineering’s Center for the Advancement of Scholarship in Engineering Education, the Center for Engineering Learning and Teaching at the University of Washington, and the Center for Design Research at Stanford University and the LIFE (Learning in Informal and Formal Environments) Center. I would also like to thank Robin Adams, Cindy Atman, Michael Knapp, Mary Beth Foglia, Laurie McCarthy, Tom Satwicz, Sheri Sheppard, Reed Stevens, Richard Storch, Kai Strunz and Jennifer Turns for their suggestions throughout the design, administration and analysis phases of the studies, and Tracie Rickert and Aki Sakamoto for their assistance with the data analysis. I would like to thank the engineering educators who partnered with me by allowing me to interview their students, and I would especially like to thank all of the engineering students who participated in the study. Bibliography ABET (2003). Accreditation Board for Engineering and Technology, Criteria for Accrediting Programs in Engineering, Baltimore, MA: ABET, Inc. Cardella, M. E. (2006). Engineering Mathematics: an Investigation of Students’ Mathematical Thinking from a Cognitive Engineering Perspective, Doctoral dissertation, University of Washington. Cardella, M. E. & Atman, C. J. (2004). “A Qualitative Study of the Role of Mathematics in Engineering Capstone Design Projects.” Proceedings of the 2004 International Conference on Engineering Education, ICEE-2004, Gainesville, FL. Coffey A. & P. Atkinson (1996). “Chapter 2: Concepts and Coding.” In Making Sense of Qualitative Data. Thousand Oaks, CA: Sage Publications. Dick, B. (2004). “Grounded theory: a thumbnail sketch,” 2002, http://www.scu.edu.au/schools/gcm/ar/arp/grounded.html, (accessed November 24, 2004).

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