This document discusses how to solve a related rates problem involving the shrinking radius of a circle. It provides the steps to determine the rate at which the area of the circle is changing when the radius is shrinking at 1 meter per second and the radius is currently 2.5 meters. The rate of change of the area is calculated to be 5π square meters per second using derivatives and the formulas for the area of a circle.
The Global Positioning System (GPS) is a network of dozens of satellites that hover out in space with the purpose of allowing people to identify their location on earth. Signals from the GPS satellites are transmitted to a GPS receiver on earth’s surface to pinpoint the satellite’s location in space. With knowledge of the satellite’s orbit and utilizing time information, a GPS receiver is able to determine its own location under the condition that four satellites are within range. However, due to the inaccuracy of the receiver’s clock when utilizing commercial GPS units for low cost, the distances calculated, called pseudo-ranges, are not accurate. Ideally, these four pseudo-ranges should intersect at a single point for a true receiver-satellite distance, but the unsynchronized clocks prevent this. To accurately determine a location, a few algebraic computations are necessary to make the adjustment for the imperfect information. These algebraic computations consist of deriving, implementing, and testing two algorithms, the Gradient Descent and Gauss Newton algorithms. Throughout this project, we will be exploring how these algorithms contribute to resolving the clock error when determining the true pseudo- range under noiseless conditions.
IOSR Journal of Applied Physics (IOSR-JAP) is an open access international journal that provides rapid publication (within a month) of articles in all areas of physics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in applied physics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This is a revision of my second order reaction rate presentation. It takes into consideration the fact that the product is formed at half the rate of disappearance of the reactant.
The Global Positioning System (GPS) is a network of dozens of satellites that hover out in space with the purpose of allowing people to identify their location on earth. Signals from the GPS satellites are transmitted to a GPS receiver on earth’s surface to pinpoint the satellite’s location in space. With knowledge of the satellite’s orbit and utilizing time information, a GPS receiver is able to determine its own location under the condition that four satellites are within range. However, due to the inaccuracy of the receiver’s clock when utilizing commercial GPS units for low cost, the distances calculated, called pseudo-ranges, are not accurate. Ideally, these four pseudo-ranges should intersect at a single point for a true receiver-satellite distance, but the unsynchronized clocks prevent this. To accurately determine a location, a few algebraic computations are necessary to make the adjustment for the imperfect information. These algebraic computations consist of deriving, implementing, and testing two algorithms, the Gradient Descent and Gauss Newton algorithms. Throughout this project, we will be exploring how these algorithms contribute to resolving the clock error when determining the true pseudo- range under noiseless conditions.
IOSR Journal of Applied Physics (IOSR-JAP) is an open access international journal that provides rapid publication (within a month) of articles in all areas of physics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in applied physics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This is a revision of my second order reaction rate presentation. It takes into consideration the fact that the product is formed at half the rate of disappearance of the reactant.
Esta es una presentacion que hice con motivo de los requisitos que exige la maestria en fisica en la Unviersidad de Bishops, en Quebec, Canada. Durante mi presentacion, hicieron incapie en un error de subindices durante el desarrollo de las ecuaciones de las ondas gravitacionales. Lamentablemente no recuerdo en que diapositiva me marcaron el error, asi que es un desafio para cualquiera que encuentre mi presentacion interesante para ser utilizada en algun proyecto. Gracias.
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
Skip The Line PowerPoint for Smarthpitch 2015roderick keys
Undergrad Technologies product is a software application and website that allows customers to bypass or reduce waiting times at tourist attractions in New York City. This is achieved by various means through the use of data analytics and technologies, partnerships with tours locations and “Skip The Line” Users.
Based in San Francisco, Dwayne Jones guides Urban Equity, LLC, as founder and managing partner. Dwayne Jones is also the founder of the nonprofit Urban Ed Academy, where he and his team work with at-risk African American, Pacific Islander, and Latino boys in the San Francisco area.
Skip The Line PowerPoint for Smarthpitch 2015roderick keys
Undergrad Technologies product is a software application and website that allows customers to bypass or reduce waiting times at tourist attractions in New York City. This is achieved by various means through the use of data analytics and technologies, partnerships with tours locations and “Skip The Line” Users.
Esta es una presentacion que hice con motivo de los requisitos que exige la maestria en fisica en la Unviersidad de Bishops, en Quebec, Canada. Durante mi presentacion, hicieron incapie en un error de subindices durante el desarrollo de las ecuaciones de las ondas gravitacionales. Lamentablemente no recuerdo en que diapositiva me marcaron el error, asi que es un desafio para cualquiera que encuentre mi presentacion interesante para ser utilizada en algun proyecto. Gracias.
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
Skip The Line PowerPoint for Smarthpitch 2015roderick keys
Undergrad Technologies product is a software application and website that allows customers to bypass or reduce waiting times at tourist attractions in New York City. This is achieved by various means through the use of data analytics and technologies, partnerships with tours locations and “Skip The Line” Users.
Based in San Francisco, Dwayne Jones guides Urban Equity, LLC, as founder and managing partner. Dwayne Jones is also the founder of the nonprofit Urban Ed Academy, where he and his team work with at-risk African American, Pacific Islander, and Latino boys in the San Francisco area.
Skip The Line PowerPoint for Smarthpitch 2015roderick keys
Undergrad Technologies product is a software application and website that allows customers to bypass or reduce waiting times at tourist attractions in New York City. This is achieved by various means through the use of data analytics and technologies, partnerships with tours locations and “Skip The Line” Users.
Dwayne Jones, president of RDJ Enterprises in San Francisco, is an experienced community development professional and entrepreneur. With a background in both public policy and nonprofit work, Dwayne Jones has worked extensively in the San Francisco and Oakland areas to establish community resources, such as Urban Ed Academy.
For the last four years, Dwayne Jones has served as president of RDJ Enterprises in San Francisco, California. Among his accomplishments with the firm, Dwayne Jones partnered with the San Francisco Public Utilities Commission (SFPUC) to write their Community Benefits Policy in January 2011.
Finite-difference modeling, accuracy, and boundary conditions- Arthur Weglein...Arthur Weglein
This short report gives a brief review on the finite difference modeling method used in MOSRP
and its boundary conditions as a preparation for the Green’s theorem RTM. The first
part gives the finite difference formulae we used and the second part describes the implemented
boundary conditions. The last part, using two examples, points out some impacts of the accuracy
of source fields on the results of modeling.
Finite element method analytical understanding was implemen ted to simulate the 1-dimensional break up model fot a jet spray . Programming was done in MATLAB
1. Below is an animation of a shrinking circle. The radius shrinks from a radius
of 5 to 1. Use the buttons below to play the animation.
A typical related rates question would be along the lines of, “If the radius of
a circle is shrinking at 1 meter per second, how fast is the area of the circle
shrinking at the moment when the radius is 2.5?”. In order to answer the
question, we must first understand the question. Below we have a diagram of
what is occurring:
1
2. Circle just before the moment of interest
Circle just after the moment of interest
How fast is the circle’s area changing from blue to green?
Or, how much area was lost when the circle changed its radius in the next moment?
So we have the following:
A(r) = πr2
A(2.5) = π(2.5)2
A(2.5) = 6.25π
(1)
This gives us the area of the circle when the radius is 2.5 (represented by the
blue graph above). We want to know how fast the area is changing from blue
to green, so we would need to compute the area of the green circle as well,
but we do not know the radius of this new circle. We have to use calculus and
that means using derivatives, since we are talking about how fast things change.
Hence we have:
A(r) = πr2
dA
dt
= π
d
dt
(r2
)
dA
dt
= π2r
dr
dt
dA
dt
= 2π(2.5)(1)
dA
dt
= 5π
(2)
You see, this is entirely a different question. The first one answers the question,
“What is the area when the radius is 2.5 meters?” It makes no use of the
2
3. information about how fast the radius is changing. The second one answers,
“How fast is the area changing...?” It does make use of the information about
the rate of the change of the radius. These are very different questions and very
different methodologies are needed to answer the questions. It is important
that we understand what the question is asking and what the symbols used in
calculus identify the quantities. Here is a listing of what the symbols mean for
this particular example:
• A(r) Area as a function of radius r.
•
dA
dt
the rate of the change of the area with respect to time, t.
•
dr
dt
the rate of the change of the radius with respect to time, t.
Translating words into the mathematical shorthand can be very helpful in using
the information as well as knowing what equation you would use to answer the
question. If we consider our question we may translate into the symbols and see
what we need. We have the following:
If the radius of a circle is shrinking at 1 meter per second, how fast is the area
of the circle shrinking at the moment when the radius is 2.5 ?
The part in red gives information about the rate of change for the radius, so we
have
dr
dt
= 1. The part in blue is the question, and it is asking about the rate
of change for the area, so we have
dA
dt
=?. The green part gives us an exact
moment of interest, so r = 2.5 in the formula for
dA
dt
. Since r has a rate of
change with respect to t, we must assume in general that it is a function of t,
hence r = r(t). So our translation comes to the following
A(r) = πr2
(t)
dA
dt
= π
d
dt
r2
(t)
dA
dt
= π 2r
dr
dt
from the general power rule
dA
dt
= 2πr
dr
dt
(3)
Now that we have established a general relationship, we may determine specific
values when we have specific information. In other words, we have a model.
Now using the specific information for this particular problem we have
dA
dt
= 2π(2.5)(1)
dA
dt
= 5π
(4)
3
4. Another way to think about this question is how much area is in black between
the circles from one instant to the next, since this is the amount of area we lose.
If we know the radius at the next moment we would have an idea of how much
area was lost. If we use the information in the problem and measure the area
at the next second we have the radius to be 1.5 meters since the radius shrinks
at 1 meter per second. We could do the following:
A(r) = πr2
A(2.5) = π(2.5)2
A(2.5) = 6.25π
A(1.5) = π(1.5)2
A(1.5) = 2.25π
A(2.5) − A(1.5)
1
= 6.25π − 2.25π = 4π
(5)
This gives the amount of area lost traveling from a radius of 2.5 meters to 1.5
meters. As we can see the actual area lost traveling in one whole second is
different than the rate of change at the instant the radius is 2.5 meters. But
using this methodology, using values of r that are closer to 2.5 meters can give
us a better approximation. The underlying assumption here would be that the
rate of change for the radius is a constant 1 meter per second. We would have
the following table
r 2.49 2.499 2.4999 2.49999
Change in Area 4.9π 4.99π 4.999π 4.9999π
As we can see, as the radius draws closer to the moment of interest, namely 2.5
meters, the change in the areas between the circles approaches the 5π value. In
essence, we have used the definition of derivative directly to build this table.
4