Digital Commons Network^™

Pasteurization Of Milk, Alejandro Barbosa Alzate

Undergraduate Journal of Mathematical Modeling: One + Two

We determined the time it takes milk, which is heated from 60°F to 150°F, to achieve pasteurization and compared this result with the standard pasteurization process of heating milk at a constant temperature of 150°F for 30 minutes. Instead of directly quantifying the bacterium population, we considered the bacteria to milk concentration ratio. To solve for the unknown time, we equated the final bacterium concentration ratio achieved through both varied temperature and constant temperature. After equating the final pasteurization concentrations we were unable to find an analytical solution, so we used numerical techniques to find the unknown heating time.

The Sir Epidemiology Model In Predicting Herd Immunity, Joanna Nicho

Undergraduate Journal of Mathematical Modeling: One + Two

The Simple Epidemic Model uses three states to describe the spread of an infection: the susceptible (S), the infected (I), and the recovered (R). This model follows the trend of an infection over time and can predict whether an infection will spread. Using this model, epidemiologists may calculate the percentage of the population that needs to be vaccinated in order to provide a population immunity from a disease. This study will compare the vaccination percentage required for herd immunity to measles, mumps, and rubella against the current percentage of vaccinated individuals.

Optimization Of A Chemical Reaction Train, Bahar Sansar

Undergraduate Journal of Mathematical Modeling: One + Two

This project consists of the optimization of a chemical reactor train. The reactor considered here is the continuous stirred tank reactor (CSTR), one of the reactor models used in engineering. Given the design equation for the CSTR and the cost function for a reactor, the following values are determined; the optimum number of reactors in the reaction train, the volume of each reactor and the total cost.

Finding The Area Of An Irregularly Shaped Room, Justin Walls

Undergraduate Journal of Mathematical Modeling: One + Two

This experiment utilized integration in a practical sense by determining the floor area of an irregularly shaped room. By taking the integral summation of three sections formed from semi-circles, the desired areas as well as positive results were found. The final data determined that the floor area of the irregularly shaped room was around 580 square feet.

Calculating Optimal Inventory Size, Ruby Perez

Undergraduate Journal of Mathematical Modeling: One + Two

The purpose of the project is to find the optimal value for the Economic Order Quantity Model and then use a lean manufacturing Kanban equation to find a numeric value that will minimize the total cost and the inventory size.

Embedded Systems - Missile Detection/Interception, Luis Cintron

Undergraduate Journal of Mathematical Modeling: One + Two

Missile defense systems are often related to major military resources aimed at shielding a specific region from incoming attacks. They are intended to detect, track, intercept, and destruct incoming enemy missiles. These systems vary in cost, efficiency, dependability, and technology. In present times, the possession of these types of systems is associated with large capacity military countries. Demonstrated here are the mathematical techniques behind missile systems which calculate trajectories of incoming missiles and potential intercept positions after initial missile detection. This procedure involved the use of vector-valued functions, systems of equations, and knowledge of projectile motion concepts.

Calculating The Time Constant Of An Rc Circuit, Sean Dunford

Undergraduate Journal of Mathematical Modeling: One + Two

In this experiment, a capacitor was charged to its full capacitance then discharged through a resistor. By timing how long it took the capacitor to fully discharge through the resistor, we can determine the RC time constant using calculus.

Social Network Gaming Trends, Michael Gathwright

Undergraduate Journal of Mathematical Modeling: One + Two

The purpose of this project was to determine how long the social network game Scratch-Offs, created by game development company Spice Rack Media, will remain financially viable. The game Scratch-Offs is a freeware game (users pay nothing for the actual software) and is funded through micro transactions (users must pay small amounts of money to play actual games). This implies a relationship between total games played and revenue earned. Using data provided by Spice Rack, we were able to develop an exponential equation that accurately depicts usage trends over time. This equation was used to determine the date Scratch-Offs will …

The Progression Of Sequential Reactions, Jack Mcgeachy

Undergraduate Journal of Mathematical Modeling: One + Two

Sequential reactions consist of linked reactions in which the product of the first reaction becomes the substrate of a second reaction. Sequential reactions occur in industrially important processes, such as the chlorination of methane. A generalized series of three sequential reactions was analyzed in order to determine the times at which each chemical species reaches its maximum. To determine the concentration of each species as a function of time, the differential rate laws for each species were solved. The solution of each gave the concentration curve of the chemical species. The concentration curves of species A₁ and A_{2 …}

Quantum Mechanical Example Of Anti-Tunneling, Tim Mitchell

Undergraduate Journal of Mathematical Modeling: One + Two

The probability of a particle being reflected by a semi-infinite energy barrier was solved using Schrödinger's equations that describe the quantum mechanical behavior of the particle. The continuity of the functions at the origin allowed the wave functions and their respective derivatives to be equated. Solution of the resulting system yielded values A₁ and B₁ which were used to calculate the reflection coefficient, (B₁/A₁)². The conclusions obtained from these results stated that the particle would be reflected by the barrier when E

Effect Of Tire Pressure On Efficiency, Miles Mullins

Undergraduate Journal of Mathematical Modeling: One + Two

Many people ignore tire pressure in their day-to-day lives. In a country striving for maximum efficiency, neglecting to maintain correct tire pressure can noticeably affect the amount of horsepower required to overcome the drag forces due to tire/roadway friction. In order to quantify these horsepower changes, 21 test trials were conducted by allowing a test vehicle to coast from 50 mph down to 20 mph at pressures ranging from 50 psi to 20 psi. After compiling the results, it was shown that lower tire pressures increased the horsepower necessary to propel the automobile.

Locomotion Of A Running Quadruped Robotic System, Bradley Savon

Undergraduate Journal of Mathematical Modeling: One + Two

Very little is known about the locomotion of quadruped robotic systems. In Latin, the term quadruped means 'having four feet'. The robotic system analyzed in this project was a software simulation program modeled after a horse. Simulation data was recorded to resolve the relationship between foot position and impulses required to accelerate or to decelerate the system while trotting. These sets of data were then analyzed in MATLAB to produce graphical representations which allowed for a better understanding of the robot's motion in the simulation environment. A function was fit to the graph of discrete data values measured in the …

Power Consumption Of A Mosfet, Frederick Selkey

Undergraduate Journal of Mathematical Modeling: One + Two

A MOSFET is defined as metal oxide semiconductor field effect transistor. These electrical components are combined or integrated to form control and logic functions for laptop and desktop computers, power controls in printing devices, motor controls and are used in many other electrical circuits. All electrical devices consume electrical power based on current and voltage. For this paper we calculated the power consumption of a Toshiba 2SK3563 MOSFET during its triode mode by finding the area under the current-voltage characteristic curve.

Canal Lock Displacement, Rick Blanton

Undergraduate Journal of Mathematical Modeling: One + Two

In this project we determine the time needed for a pump to raise the water level in a canal lock in order for a boat to continue upstream. Using calculus methods and elementary physics, it was determined that it would take roughly 5 minutes for a single 60 horsepower pump to raise the water level the required 10 feet. We conclude that the lock is fairly efficient but offer some suggestions to increase the time efficiency of the lock system.

Brazilian Pepper's Impact On Barrier Island Biodiversity, Grayson Mccormick

Undergraduate Journal of Mathematical Modeling: One + Two

The Brazilian pepper is an invasive plant responsible for destruction of habitats of species native to Florida. We attempted to measure the impact of the Brazilian pepper on native plants by studying how it affects the biodiversity of Honeymoon Island, one of Florida's barrier islands. For this purpose we determined the population sizes of the Brazilian pepper along with five other species of plants on two quadrats of land laid out in Honeymoon Island State Park. Both Simpson's Index of Diversity and Shannon-Wiener Index were used to estimate the biodiversity and assess the impact of the Brazilian pepper on the …

Automated Determination Of A Package's Center Of Mass, Ayaz Hemani

Undergraduate Journal of Mathematical Modeling: One + Two

In order to address the issue of increased efficiency and better planning for parcel shipments, an automated computer program was developed in Microsoft Excel that calculates center of mass and moments of mass with greater speed and reliability than currently implemented systems. This simple program requires only a variable density function and limits of integration for a given object as input within the spreadsheet system. Once the required input has been provided, a series of chain calculations, with the help of a Visual Basic for Applications (VBA) script, is able to process the input, which is done through integration and …

Pollination Of A Canary Tree Flower, Abbie O'Hern Weeks

Undergraduate Journal of Mathematical Modeling: One + Two

Pollination is an essential part of the life cycle of flowering plants. We perform an experiment to determine how long a canary tree flower is accessible to pollinating insects so that fertilization can take place. We conducted an observational study where we measured the size of the same flower and charted its growth each day. With the observational data we constructed a scatter plot and from the graph we fit a cubic function to the data. We conclude that in the lifespan of a canary tree flower, pollination begins at 5 1/2 days and ends approximately 9 days later.

Population Dynamics Of Free-Roaming Cats In Florida's Lee County, Benjamin Taylor

Undergraduate Journal of Mathematical Modeling: One + Two

We investigate whether the Trap-Neuter-Return (TNR) program can be effectively used to control the population of free-roaming cats in Florida's Lee County. We do this by estimating the number of cats that must be spayed/neutered in order to keep the population from increasing.

Dosimetry Calculations Of Radiopharmaceuticals, Kelbe Jahnke

Undergraduate Journal of Mathematical Modeling: One + Two

The goal of this project is to determine the absorbed dose of radiation to a patient using ^99mTc-MAA as a tracer for a lung scan using the MIRD formula accounting only for the radiation dose contributed by the target organ. The radiation dose for other organs to the target organ is considered negligible as the uptake of ^99mTc-MAA from other organs is less than 1%. In addition to calculating the absorbed dose of radiation, the initial dose rate was also calculated by back solving to determine the total mass of the lungs. The radiation dose to the patient …

Inverse Kinematics, Joel Sereno

Undergraduate Journal of Mathematical Modeling: One + Two

Inverse kinematics is the process of converting a Cartesian point in space into a set of joint angles to more efficiently move the end effector of a robot to a desired orientation. This project investigates the inverse kinematics of a robotic hand with fingers under various scenarios. Assuming the parameters of a provided robot, a general equation for the end effector point was calculated and used to plot the region of space that it can reach. Further, the benefits obtained from the addition of a prismatic joint versus an extra variable angle joint were considered. The results confirmed that having …

Motion Of A Pendulum, Jared Wynn

Undergraduate Journal of Mathematical Modeling: One + Two

The objective of this project is to derive and solve the equation of motion for a pendulum swinging at small angles in one dimension. The pendulum may be either a simple pendulum like a ball hanging from a string or a physical pendulum like a pendulum on a clock. For simplicity, we only considered small rotational angles so that the equation of motion becomes a harmonic oscillator.

Light Attenuation At Molasses Reef, Ashley Earls

Undergraduate Journal of Mathematical Modeling: One + Two

Estimating the amount of light available at different depths of the ocean is important for gaining a better understanding of coral reefs. It is especially useful to be able to get such estimates without having to perform direct measurements. Although accurate, the direct measurements are costly, time consuming, and usually limited to relatively small areas of interest.

One statistic that could be useful for estimating the amount of available light is the rate at which it declines with depth (the rate of light attenuation). In this project, this rate is calculated for four different wavelengths using data for 30m water …

Volume Of An Industrial Autoclave, Nicholas Madaffari

Undergraduate Journal of Mathematical Modeling: One + Two

We were able to determine the volume of an industrial autoclave sterilization tank using a technique learned in calculus. By measuring the dimensions of the tank and roughly estimating the equation of curvature at the ends of the tank, we were able to revolve half of the end of the tank around the x axis to get its fluid volume. Adding the two volumes of the ends and the volume of the cylindrical portion on the tank yielded the total volume.

Maximum Power From A Solar Panel, Michael Miller

Undergraduate Journal of Mathematical Modeling: One + Two

Solar energy has become a promising alternative to conventional fossil fuel sources. Solar panels are used to collect solar radiation and convert it into electricity. One of the techniques used to maximize the effectiveness of this energy alternative is to maximize the power output of the solar collector. In this project the maximum power is calculated by determining the voltage and the current of maximum power. These quantities are determined by finding the maximum value for the equation for power using differentiation. After the maximum values are found for each time of day, each individual quantity, voltage of maximum power, …

Escape Velocity, Nikola Vlacic

Undergraduate Journal of Mathematical Modeling: One + Two

In this project, we investigated if it is feasible for a single staged rocket with constant thrust to attain escape velocity. We derived an equation for the velocity and position of a single staged rocket that launches vertically. From this equation, we determined if an ideal model of a rocket is able to reach escape velocity.

Optimal Location Of An Oil Storage Facility, Giovanni Quiel

Undergraduate Journal of Mathematical Modeling: One + Two

Given three oil drilling sites, we devise a method to determine the optimal location for a storage facility such that the total length of pipeline required to connect each site to the facility is minimized. First we represent the total distance of piping as the sum of the individual distances between the storage facility and each oil well. From this, both the optimal overall pipeline length and the location of the storage facility which realizes the minimal length can be determined through the first and second partial derivative tests.

Total Number Of Synapses In The Adult Human Neocortex, Thai Nguyen

Undergraduate Journal of Mathematical Modeling: One + Two

The brain is composed of glial cells and neurons where synapses form connections between neurons and other cells. Since synapses are very small, so either a light or electron microscope is required to see them. Unlike other mammals, synapses in the human brain deteriorate rapidly upon death making them difficult to study. This project constructs a simple model for the number of synapses in the human neocortex by age and sex based on the amount of neurons. This hypothetical model can also be used to study the impact of Alzheimer's disease and other forms of dementia that are marked by …

Stormwater Management System Drawdown, Ahmad Chehab

Undergraduate Journal of Mathematical Modeling: One + Two

This project concerns the computations required to determine the drawdown for the retention/detention of ponds. Drawdown refers to the volume of water in a pond that decreases as the water flows out. The falling head equation has many applications and can be used to calculate the drawdown of a pond through various shaped openings. In particular, we analyze four outflow structures: a rectangular-notch weir, a v-notch weir, a round orifice, and an underdrain. For each instance, we modify the falling head equation to reflect the shape of the respective orifice.

Radiographic Imagery Of A Variable Density 3d Object, Justin Stottlemyer

Undergraduate Journal of Mathematical Modeling: One + Two

The purpose of this project is to develop a mathematical model to study 4D (three spatial dimensions plus density) shapes using 3D projections. In the model, the projection is represented as a function that can be applied to data produced by a radiation detector. The projection is visualized as a three-dimensional graph where x and y coordinates represent position and the z coordinate corresponds to the object's density and thickness. Contour plots of such 3D graphs can be used to construct traditional 2D radiographic images.

Nerve Cell Deterioration Associated With Alzheimer's Disease, Yaping Tu

Undergraduate Journal of Mathematical Modeling: One + Two

Alzheimer's disease is an extremely serious condition that is challenging to diagnose. We have used experimental data to compare the rate of decay of entorhinal cortex (EC) neurons in various stages of Alzheimer's. We observed that the rate of EC neuron decay in the patients without Alzheimer's is close to zero, linear in mild cases, and quadratic in severe cases. We believe that described estimates may help to diagnose the disease as well as its stage.