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    Modeling Scenario
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    1-032-WordPropagation-ModelingScenario
    This activity is a gentle introduction to modeling via differential equations. The students will learn about exponential growth by modeling the rate at which the word jumbo has propagated through English language texts over time.
    Googleexponential growthlanguageword problemsetymologyNgrampropagationJumbo
    Modeling Scenario
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    1-035-DotseroVolcanoEruption-ModelingScenario
    In this scenario, we use Carbon-14 dating of the Dotsero volcano in Colorado as a way of emphasizing this multistage process of modeling.
    exponential decayhalf-lifedecayEuler's methodcarbon datingdecay rateCarbon-14Dotsero volcanovolcano
    Modeling Scenario
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    6-025-WhalesAndKrill-ModelingScenario
    Students will use Excel to observe qualitative behavior in a simulation of a predator-prey model, with blue whales and krill as the predator and prey populations, respectively.
    krillwhalepredatpr-prey
    Modeling Scenario
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    1-108-PoissonProcess-ModelingScenario
    In this project students learn to derive the probability density function (pdf) of the Poisson distribution and the cumulative distribution (cdf) of the waiting time. They will use them to solve problems in stochastic processes.
    Probabilitysstochastic processesPoisson processwaiting timearrival
    Modeling Scenario
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    1-102C-CancerGrowth-ModelingScenario
    This module guides students in the use of differential equation models to predict cancer growth and study treatment outcomes. Several classical models for cancer growth are presented including exponential, power law, Bertalanffy, logistic, and...
    optimizationcancerlogisticexponentialpredictiontumorgrowthdata fittingpower lawBertanffyGompertz
    Modeling Scenario
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    6-068-VisualizingPredator-PreyCycles
    In this modeling scenario, we propose a gentle introduction to limit cycles using nullcline analysis and a spreadsheet graphical approach via the fourth-order Runge-Kutta method. We will be offering three predator-prey models, each more complex...
    nullclinespredator-preyGeometrypredatorRunge-Kuttapreylimit cycle
    Modeling Scenario
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    1-102-CancerTumor-ModelingScenario
    This module guides students in the use of differential equation models to predict cancer growth and optimize treatment outcomes. Several classical models for cancer growth are studied, including exponential, power law, Bertalanffy, logistic, and...
    optimizationlogisticpredictiondata fittingpower lawGompertzcancer growtheponentialBertalanaffy
    Modeling Scenario
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    5-080-SpaceFlightRecolonize-ModelingScenario
    This project is a combination of differential equations, multi-variable calculus, and vector calculus with use of technology to model colonization of a new planet. Students solve a system of second order differential equations to model a planet
    optimizationsum of square errorsvector calculusLagrange multipliersline integrals
    Modeling Scenario
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    9-010-TravelingWave-ModelingScenario
    Students are taken through a traveling wave analysis of a porous medium model. While the starting point is a nonlinear partial differential equation model, after a change of variables, students are led quickly to an ordinary differential equation...
    traveling wavesseepage velocityporous medium
    Modeling Scenario
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    1-052-SaltWaterTanks-ModelingScenario
    We offer three mixing problems, of increasing order of difficulty, in which salt is coming into a tank of water and upon instantaneous mixing is leaving the tank.
    salttankmixing
    Modeling Scenario
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    4-065-GasInjection-ModelingScenario
    Students use programs (or create their own code) based on exponential box-scheme approximations for solving systems of nonlinear differential equations that contain small parameters for the highest derivative terms or singularities in boundary...
    Prandtl-Blasius equationsboundary layergas injectionesponential box schemeuniform convergenceFORTRAN
    Modeling Scenario
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    9-015-UnearthingTruth-ModelingScenario
    This project introduces \textit{electrical resistivity tomography}, a technique of interest for geophysical imaging, used to produce images of underground features or structures by using electrical current.
    geophysicalImagingStatistical inverse problemtunnelconductionHolocaustVilniusDivergence TheoremfluxLaplace's equationimpedanceelectrical resistivity tomographytomographyresistivity
    Modeling Scenario
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    3-051-ProjectileMotions-ModelingScenario
    We consider several instances of projectile flight without resistance, one on level ground and one from edge of cliff to determine maximum distance and placement.
    trajectorylaunching radiantprojectile motionprojectileno resistanceangle of launch
    Modeling Scenario
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    3-052-OptimalProjectileFiring-ModelingScenario
    We offer the opportunity to model a projectile's trajectory in several cases, all without resistance.
    projectile motionprojectilelaunch anglemaximum range
    Modeling Scenario
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    1-144-HeatCool-ModelingScenario
    We offer an opportunity to build a mathematical model using Newton's Law of Cooling for a closed plastic baggie of liquid inside a liquid container.
    temperatureNewton's Law of Coolingcooling
    Modeling Scenario
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    1-131-CaffeineElimination-ModelingScenario
    We model the concentration of caffeine eliminated from the human body at a rate proportional to the concentration. This is a ``first-order reaction'' in the language of pharmacokinetics -- the study of how drugs move in the body.
    caffeineeliminationpharmacokineticsdrug
    Modeling Scenario
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    1-078-MonodGrowthModel-ModelingScenario
    Students model growth of bacteria E. coli in a limiting nutrient environment using data from a historical study.
    bacterianutrient cyclesgrowthMonodconsumption
    Modeling Scenario
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    3-033-S-TimeUpTimeDown-ModelingScenario
    We seek to compare for the time a projectile takes to go vertically up with the time it takes to return to its starting position.
    projectile motionapexvertical motion
    Modeling Scenario
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    5-024-PhGreatLakes
    In this teaching modeling scenario, we demonstrate how lessons on salt-tank compartmental modeling can be used to predict phosphorus levels in the Great Lakes.
    Great Lakeslimnologycomparmentpollutioncompartmental modelsphosphorous
    Modeling Scenario
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    1-015-Torricelli-ModelingScenario
    We help students develop a model (Torricelli's Law) for the height of a falling column of water with a small hole in the container at the bottom of the column of water through which water exits the column.
    dataSpanishtankcolumn of waterapertureTorricelli's LawConservation of EnergytSolverwater flowExcel Solver