It seems to me that every year there is new, incredibly ground breaking technology. Math included. This technology always is portrayed as the "future." Its gonna change the world around us! Well, I believe that we do that by ourselves. We already have the technologies to change our world, how we use should be the most pressing issue.
Currently, everything we create is built in order to streamline the process. As shown in the article, shopping is included. The fact that a company has the rights to what music I'll be buying next, or my favorite author may, in fact, be quite frightening. I am a believer in the fact that the more this technology piles on, the more cautious we need to be about how we use it. Computers already have capabilities far beyond that of any human. As a race, we need to hold true to our roots. We must embrace technology in a way that we do not inhibit our natural tendencies.
I don't mean to be all superficial and such, but is this really that far from possible?...
Alright, maybe thats a little far fetched. But, I would say that we are likely on our way there.
I think that all this technology is definitely beneficial and could greatly change the world we live in. It has the opportunity to benefit all of us. However, I would stress that we should keep in mind where we would like to be, and where we are going.
We’ve discussed the learning paradigm of “Watch it, Do it, Teach it”. It is one thing to watch someone do something; still another to do it yourself; but quite another to teach it to someone else. Real learning comes from teaching. Learning also does not occur in a vacuum. We learn through our interactions with others and with our environment. Collaboration is an important life-skill for the 21st century global citizen.
The goal of this final project is to work together to teach a calculus concept
and by doing so learn at the same time.
Nowhere is it more evident that calculus is the mathematics of change than in related rates problems. When two quantities change over time, the rates of their change are often related to each other. That’s one of the differences between algebra and calculus. In algebra we study the relationship between quantities. In calculus we study the rates of change of those quantities. In this respect, an algebraic equation is a static snapshot of two quantities frozen in time, while a differential equation of calculus represents a dynamic video of those quantities set in motion.
Within each group, the final project consists of three tasks:
PRESENT a related rates problem. Select a scenario from the pool of classic related rates problems or create an original situation with similar parameters. Act out the scenario either by video or by stitching together a storyline of still photos. Be creative! Use toys, props, or other inventive ways to illustrate your scenario (puppets, claymation, etc.). Here are some examples to spark your imagination. Filling a cylinder Baseball runner Moving sea creatures Stone thrown in lake Flying pig Reeling in the big fish
Your related rates problem should be presented clearly and concisely; i.e. in a way that it could be used as a homework or test problem. Make sure that it is fully-specified, but not redundant (that is, there can’t be too little or too much information to solve the problem). Your problem should not be ridiculously difficult, nor should it be so simple that it is practically trivial. It must require calculus to solve.
SOLVE your related rates problem. The solution will include a written description of the problem. All variables used to model the scenario will be clearly defined. The solution will contain a graphic to illustrate a “snapshot” of the scenario; and each step of the solution will be properly annotated.
REFLECT on your work on this project.
Each member of the group will describe his/her contribution to the project.
Each member of the group will write a reflection on what he/she learned from working on the final project. This may include the use of a new Web 2.0 tool or computer skill, thoughts on collaborative group work, or insight into related rates problems.
The presentation of your problem will be posted to our blog. It will begin with an introduction which includes a statement of purpose, the nom de plume of each group member, and a brief description of the project scenario. After the presentation, provide a hyperlink to our wiki where you will post the solution and reflection portions of the project. Here is the rubric for the project.
The finished project is due by 3:00 pm on Friday, March 5. This final project will be a reflection of your learning, group effort, and commitment to excellence. I hope that you have fun teaching the concept of related rates to others who will benefit from your knowledge.
“There has never been a good time to be a mathematician.”
In today’s society, everything falls on math. All the careers have some math concept involved in somehow. This has been the case for several years, but nowadays, as evolutions keeps on going farther, the need of math increases. Statistics is used in the medical field to determinate the improvement of health conditions, in business to improve the quality of work done, in security agencies to insure the safety of the citizens.
First, this article explains the pros of math and its development. The world is now dominated by Web2.0. Education, entertainment, shopping, careers, etc. everything can be done online, and this won’t be possible without the contributions of mathematicians. Since they are much needed in today’s society, the article suggests that students, especially in America, should be encouraged to pursue their careers in math or math related major fields. The example is given in the case of the NSA which employs only US citizens. From here, it can be concluded that the security of Americans is limited, since it is less than what it could be.
The other side of the article explains how these ongoing improvements in the math’s world are quite disturbing. Since people are concerned about the privacy problem, it’s hard to make improvement in medicine for example. Since much of these math based programs use people’s private information, there have been some complaints about this, and they are refusing to give up their information to the public. This pushed mathematicians, doctors and other concerned people to look for other alternatives in their problem solving strategies
Overall, this article made me think about all the connections that can be made between the world of math, which is always perfect (with exact answers), and the real world, which is always a cumulative result of more or less estimations and probabilities. There have been some times when, in the middle of class or while doing a math problem, I said to myself “when am I ever going to use this?” This article reminded me that’s always a rhetorical question I have in mind, that somewhere in the back of my conscious mind I know that I will use it. It also reminded me the saying that says “Math is related to Physics, Physics related to Chemistry, Chemistry related to Biology, and Biology related to Life.” So, in the end, math is related to our lives, either directly or indirectly.
This article mainly focuses on the fact that we live in a predominantly technology focused world which means math is becoming more apparent in our everyday lives. Whether we know it or not, every time we check our email, or watch tv, math played a big part in getting that type of information into our homes. Since computers became popular around the world, it is much easier for companies to track their products and to see what their customers really think. This type of media, I think, makes the world a more communal place since everyone gets to express their views and show them to the whole world. I think that the math the article talks about is more behind the scenes more most of the population. Many people don't know how much math goes into their daily activities, and they will never need to know or be exposed to it in any way. I think you would only really realize how big of a role math plays in our technological society if you had a job as a computer programmer, or your job entailed a lot of statistics. To use computers the average person doesn't need to know much about math, in fact it seems like each new version of various operating systems make it so there is less and less calculations and knowledge involved. Now almost anyone can use a computer. I think the world is moving towards simplicity. Only a select few will know all the math involved in shaping our society. Computer programming knowledge will be the new skilled craftsmen. Instead of needing painters and woodworkers, in the future we will need more computer programmers.
I found this video that shows some of the math involved in computer programming:
This article really made me think about my future. It seems as though, if you want to be someone with influence in the world, your best bet is to become a math major. Baker talks of how math greatly influences the world we live in. Although i did not fully understand the relationship, I understood that it math is very critical in the world today. Most everything in our daily lives can be analyzed in number because of patterns that exist. Words, movements and emotions can all be described in numbers. Its all quite amazing. Businesses use these patterns to predict what we people want in order to sell products and such. Internet, for example, is monitored by computers that gather data from keywords that we type to build a sort of profile of each person in order to see what we want. These companies use the data to advertise their products. Though this analyzation of the internet, companies can find out what is popular among people fairly easily. This is a concern to many because it is in some way an invasion of our privacy. There are debates regarding the monitoring of the internet and phones in congress. On the hand, these computers can pick up suspicious behavior that could help with the war on terrorism. Companies are working to protect privacy as well as get enough info to build good data. All these companies are at the frontier of the technological movement of mathematics playing a huge part in our daily lives. And the workers at these companies are kids straight out of college who are good at math. And they make a good amount of money if they score a big idea that another company wants to buy. Its a good place to be, which make me consider taking that route. Plus, the US needs more locals to be these math geniuses because so many of the people working at these companies are foreign born. Its also kind of scary how much math can do. How it can predict what people want and how it can pick up conversations through computers that were intended for few people. Our whole way of life is being converted into numbers. Although computers have the capability to cure illnesses and stuff in is important to note that eventually they may turn on us like they did in terminator, just a thought. In conclusion, like Baker states, it is a good time to know math because it is what is moving the world forward.
The article was about how math is used; it is used in everything now. It is used to track the way people buy products, where they buy products, and the way they react to ads. Google uses it to see which sites people are clicking on, and what sites they are buying things from. All of this monitoring, does have it’s downsides though, even though companies will be saving money what about people’s personal privacy, do they still deserve that?
I think that the main problem with this whole thing is that idea of lost privacy. I don’t necessarily want people watching all of the things that I buy, and all of the things that I click on the Internet even if it isn’t inappropriate. So as the article pointed out I would probably take measures to see that the companies couldn’t track what I said, bought and clicked on. The article also pointed out that the government is using the same data to find terrorists, and track diseases. So it is a mixed bag whether we should trust this whole database of information about us. What is interesting and more concrete though is that people who are going into college right now should probably major in something math related if they are at all into it because that is where the market is. At least that is what the article said. I disagree. I think that it would be sad if all these bright people decided to go into math simply because that is where the jobs lie. People should follow their passion. This was an interesting article but I disagree with it in some aspects, I don’t think that our lives should be so closely monitored even if it does help homeland security or our health, it isn’t natural, or healthy in my opinion. And I don’t want my life to be run by either a government or by industries.
The article directly handles the effects of a rapidly changing world and how Calculus and other even more advances areas in the field of mathematics can be used to not only describe, but predict and change our future. They quite clearly state that the need for advanced mathematics is growing nad the those who will be the most successful in the job world will be those who have atleast some portion of advanced mathematics.
How Math Transforms Industries discusses many different fields ranging from business to media and explains how each has and will be effected by further advances in technology and mathematics. How Much Math Do We Need to Know discusses just that, how much is going to be enough? Touching on Calculus, statiscs, advanced algebra and Geometry. Pointing out that in extremely advanced Geometry, techniques were used from that field to create search engines such as Google, and Yahoo. How? I don't know, but it says thats how they did it.
For me this article to me wass incredibly fasinating. The fact that we are moving into an age where quite simply understanding and being able to apply multiple types of advanced math to pieces of data,collections of information, to quite simply predict the future. Adjust plans, marketing schemes, business models, all based off of data. This following paragraph clearly illustrates a prime example of what has been done and what is still to come form this field of advanced mathematics in business.
"The clearest example of math's disruptive power is in advertising. There Google and other searchcompanies built on math are turning an industry that grew on ideas, hunches, and personalrelationships into a series of calculations. They can pull it off because, quite simply, they know wheretheir prospective customers are browsing, what they click on, and often, what they buy. Internetcompanies use this data not only to profile customers but also to pitch for more contracts. Some 18months ago, 3blue-chip companies, from Procter & Gamble Co. () to Walt Disney Co. (), underwent a series of tests promoted by the Interactive Advertising Bureau, an industry group. These studiescrunched consumer data to measure the effectiveness of advertising in a host of media. The resultscame back in hard numbers. They indicated, for example, that Ford Motor Co. () could have sold anadditional $625 million worth of trucks if it had lifted its online ad budget from 2.5% to 6% of the total. Ford responded vigorously: Last August it announced plans to move up to 30% of its billion adbudget into media targeted to individual customers, half of it through online advertising. Such movesare sure to generate even more data, giving greater clout to the numbers people."
http://www.businessweek.com/print/magazine/content/06_04/b3968001.htm?chan=gl (2 of 6)1/17/2006 2:47:50 AM
While clearly beyond my understanding of how they can make these predictions, i find it amazing that as math has become more and more complicated it has, due to advances in technology only become more and more applicable in the modern work force.
As time progresses mathematic literacy is going to become more and more relavant and I believe more and more a part of our daily lives. So while at times maths and science appear to be largely irrelavant to our futures and what we may possibly want to do with our lives, they may prove to be by far the most important pieces of information we learn in our time in school.
I'm not sure if I'm being sarcastic right now, but this article was truly interesting to me. It explained the close relationships between mathematics, numbers and our daily life containing internet.
Cover story starts with the giant title saying that math will rock the world which is totally true. It starts the article with the system that organize, and structures any written information published by its context, concept, and theme. By doing so, it says that it needs numbers and math to accomplish it. In one sense this sounds totally new to someone, but in other side, this is just a basic common sense that already has been known to the public.
Since the beginning of the construction of computer, math always has always been in people's life style. Like it stated in the article, math is essential to everyday life. Everything is organized with math equations, and everything is programed with algorithms online, and companies have been dealing with their costumers and products based on mathematical models.
Issue of privacy has to come right after that. In the several articles in the cover story it stated about how individual's privacy is becoming public by the development of technology based on math. People's lives have been monitored and gathered as a data for the bigger companies. This is one of the greatest fear that people might have. For the company they have to do this research in order to gain biggest possible profit that could be happen.
Mathematician will be the largest group of people who are ruling the world. It is true that most of the job will provide math skills. Highest rates of mathematics elites will gain the most profit. Math will rock the world; it has been and it will.
Truly, this article is boring. It talks about things that everyone knows. 21st century, for sure is a century of math, of numbers, of data.
Basically, this article is telling us how important it is for people to understand math in this century. Our society now is filled up with data. People who could read and understand these data are appealing to all the big companies. By analyzing the data, math geeks could find out tons of information beyond these numbers. These new information they investigated provides the company a better view to the market and gives the big company a big step ahead of others.
However, there is a big flaw in this new system. As the article says “The power of mathematicians to make sense of personal data and to model the behavior of individuals will inevitably continue to erode privacy”, I think we have lost our privacy for decades. There are cameras everywhere. Government could simply monitor or track someone’s phone with the title of “National Security”. Thing cannot get any worse now, since we are already at the bottom of the abyss.
The title of the article says “Math will rock your world”. I think that is incorrect. I say math is rocking our world and has been rocking our world for at least 20 years.
Think about it, if you will. There are math everywhere in our world. Since the First industrial revolution, science walked into the stage, math also became an irreplaceable part of our society. A huge part of science is actually math. Physics obviously has to do with math. Chemistry involves math. Environment science needs data. Statistics are also math. All the industries, mechanicals, company finance involve what? Math. They need math to sustain the system.
So learning math will be the tendency for the future. For sure, there are other ways to survive in this society, but math will be, and must be the most important thing to know to survive.
In the first portion of the cover story Math Will Rock Your World, Steven Baker discussed the different applications of math in our world today. The first section discussed the conversion of words into math in order to classify and organize them. He gave the example of Neal Goldman, a math entrepreneur who takes press articles and blog posts and groups them together with related pieces based on their content. The article also discussed mathematical models and their ability to predict a variety of different things such as how to become more productive, make a profit, etc. Through collecting data, they use their findings to establish trends. One potential problem with this system lies with the population and their ability to breech the system. Although this system appears very dependable it also brings its challenges for the United States and the world. As a nation we must work towards creating more mathematicians at home and prevent outsourcing by preparing our students for the business world. Another challenge math entrepreneurs face is individual privacy. They must discover a means of invading privacy to the least extent possible by trying to share group information without sacrificing that of the individual . It surprised me how much math manifests itself in our daily lives. "Some models predict what music we'll buy, others figure out which worker is best equipped for a particular job." I never realized the full potential of math in everything we do. I was also astounded by the extent to which math can invade our privacy and how frequently numbers have begun to replace the individual. The second portion of the cover story Online Extra: The NSA: Security in Numbers discussed the usage of numbers in terms of security. Baker discussed the NSA's role and how they use math to help figure out terrorist plots and other issues of security. One of the challenges the NSA faces manifests itself in recruiting its mathematicians. They are faced with competition from Google, Yahoo and other organizations. How are they managing? They offer a more quiet, stable lifestyle, which is especially more appealing to women. I thought it was interesting that they were looking for younger employees who did not know the company as well. I assumed they would want older, more experienced workers. I found the selection process quite interesting in that it is a contest which acts as somewhat of an advisory for their decisions. The third portion Online Extra: Search Advertising by the Numbers discussed bidding on keywords. Baker specifically discussed Imran Khan and his involvement in E-Loan. Kahn makes sure those who are looking for his product will surely find it. Parts of this passage made it seem like math can often get a little bit too invasive, "It (numbers) enables marketers to track customer behavior, and replaces hunches with science." It may have just been the way I read into the article, but that sounds as though it could be problematic in terms of individual privacy issues. I also found the discussion of bidding on keywords and the science behind it a bit confusing. How Much Math Do We Need to Know? summed up the different uses of Calculus, Algebra and Geometry, Statistics and Probablility, and Math Tools in different careers. Some of the associations are not as obvious as they seem, for example, advanced geometry is used not only in floor tiling but also in designing search engines. With the growing demand for mathematics based, or even just related professions, knowing your math might prove to be helpful. I guess calculus really will pay off in the long run! How Math Transforms Industries discussed the specific uses of mathematics in many different fields, such as consulting, police and marketing and the media. Who knew a company could "turn written articles into bits of geometry and organize them in a virtual library?" and eventually these automatic systems could make editors obsolete. I found it amazing how much of a difference math has a potential of making on our society. We always talk about technology taking over life as we know it, but what about math?
In the article "Quantifying Humanity" the author discusses the uses of math in the business and technological worlds. The author specifically references each math genre (calculus, geometry, statistics etc.) and gives examples of how they can improve technology and create jobs for people in any field of study. For example he talks about how statistics is a major contributer in every job market because whether you are dealing with fashion trends, environmental patterns or oil sales you will be analyzing large sets of data and comparing trends. The article also talks about the importance of calculus in technological advancements and how geometry is not just for laying tile floors but is a huge aspect of engineering.
I personally thought this article was very interesting. I already knew that math was a huge part of the future, but had no idea of the relevance it serves in our every day lives. It was very interesting to learn about the enormous variety professions that use math to succeed. I have never really thought about how math is a huge part of the sales industry or how it helps major distributers like Ford and even insurance companies. It was great for me to see all of the practical uses of math, because it reminded me of why I continue to take math classes and work hard to understand every aspect of math. Math is not my strongest subject, but I believe that it will create many opportunities for me, as well as the rest of the world, in the future. Math has definitely rocked my world, has it rocked yours?
This article talked about how math is beginning to become more and more a bigger part of our everyday lives. One way it does this for example, is using math to analyze how well advertising is doing for a company, and how changing advertising can effect profits and sales. It is not limited to this though. Essentially what this article was saying is that math can be used to model and analyze just about everything in the world. Machines, productivity, security, sales, even humans. Another thing that I found really interesting was how important this math really is. The top mathematicians in the world, a select few, are becoming some of the highest paid people. I never would of thought that math would be getting you a 6 figure salary right out of college. This just goes to show how important and how helpful these models are for companies. Another thing it talked about is how these models can be an invasion of privacy, but I didn't really see how. For the most part, I see these models as looking at numbers not people. It turns people into numbers and so I don't really see the invasion of privacy, unless people feel invaded being turned into numbers. It is really interesting though, to think that everything can be justified with numbers. This is one thing the article reminded me of was the movie Pi, an artistic, mind-blowing thriller about a paranoid mathematician who sees math everywhere and is one the verge of discovering the unifying number that explains everything. Here's a trailer:
The three points he makes about math being everywhere and thus it being able to explain the patterns everywhere in our world is pretty interesting, and relates back to this article. When I first saw this movie, I thought it was a little far-fetched, but now... I'm not so sure. The thing that really got me was that they're working on ways to model human behavior, a supposedly random and unpredictable thing. Maybe math is everywhere, maybe eventually we'll be able to see the patterns present in nature, and maybe someday we can use that knowledge of the patterns for the betterment of humanity and our world. We'll just have to wait and see.
In the cover story in Business Week, Steven Baker addressed how math is becoming necessary in today's world. Because our world is now filled with oceans of data, math is being used more and more to 'mine', interpret, and use the data. The beginning of the article talked about how mathematical models can be used by companies to increase revenue:
"These studies crunched consumer data to measure the effectiveness of advertising in a host of media. The results came back in hard numbers. They indicated, for example, that Ford Motor Co. () could have sold an additional $625 million worth of trucks if it had lifted its online ad budget from 2.5% to 6% of the total."(3).
This kind of data is hugely valuable and could be potentially used on a federal level to calculate spending in the government. A lot of the article, however, talked about how math can be used with people, "The power of mathematicians to make sense of personal data and to model the behavior of individuals will inevitably continue to erode privacy." (2). Companies can use math to quantify and analyze their consumers and use that to create effective advertising.
I feel that, if taken too far, this could lead to a bad future. If we live in a world where every person is one of several million that fit a certain profile, that will be used for advertising, our society will continue to be controlled by media and consumerism. I once read a book call Fahrenheit 451, and this article reminded me of that book. In the book, books were banned, and nobody ever had independent thoughts or opinions. Everyone was stuffed full of information and data, but never taught to think about it or draw conclusions from it. Of course, this is a huge exaggeration of our world, but I feel that if the world moves more towards machines, and numbers, and data that generalize society into numerical patterns and equations, the opportunity for free thought and imagination will become less.
As for the part about how what we're leaning now is actually going to help us in the future, I think the importance of knowing high level math is overestimated. Really, we only need a few geniuses to write the computer programs and then the computers do all the math. I guess though, when math geeks and computers rule the world, those with calculus level math education have a better chance of survival.
"The next Jonas Salk will be a mathematician, not a doctor." (3) The article 'Math Will Rock Your World' in BusinessWeek explains why.
This article is essentially an overview of one of the directions that math is going in today's world. It focuses on how applied math is becoming increasingly valuable in the world of computer sciences and virtual analysis. Today, those programmers who can write the best algorithms or mine the most crucial data from the internet and its vast quantity of 'unstructured data,' are becoming increasingly valuable. Programs that sift through thousands of blogs, articles, advertisements, etc., and can quantify their results in a way that can create profit are not only becoming more numerous but also more accurate and because of that, more valuable. Because there is such a host of information on the web, the algorithms that are able to pick out useful patterns and tendencies are becoming the new '.com's' of our world. And the people who are math savvy are being increasingly rewarded both monetarily and with power and position.
Beyond that, there was a huge emphasis on humans being transformed into points on a graph. Our behavior, race, achievements, socioeconomic status, and anything else that can be seen as an essential character brick, are being transformed into a statistic on a chart.
“The clearest example of math's disruptive power is in advertising. There Google and other search companies built on math are turning an industry that grew on ideas, hunches, and personal relationships into a series of calculations...Rising flows of data give companies the intelligence to home in on the individual customer. Internet marketers are the natural leaders, but traditional businesses are following suit. Gary W. Loveman, CEO of casino giant Harrah's Entertainment Inc. () and a former Harvard B-school professor, has led the company to build individual profiles of millions of Harrah's customers. The models include gamblers' ages, gender, and Zip codes, as well as the amount of time they spent gambling and how much they won or lost. These data enable Harrah's to study gambling through a host of variables and to target individuals with offers, from getaway weekends to gourmet dining, calculated to maximize returns. In the last five years, Harrah's has averaged 22% annual growth, and its stock has nearly tripled...[Another company scans the web for articles and blogs and] breaks down English messages into the smallest components -- words, phrases, grammar, even emotions -- and turns them into math.” (3-4)
Essentially, that which has been reserved for the qualitative side of analysis is now being analyzed quantitatively.
When I first read this, I was somewhat disturbed. "What right does some algorithm have to turn me into math?! What a gross violation of my humanity!," I thought. But then I realized that if these companies are finding ways to successfully analyze things that have so long been considered subjective, then maybe this isn't a case of the proverbial David getting squashed by Goliath. It's not as if these programmers are making things up about me, they are doing nothing other than analyzing what is already there. And what's so horrible about that? Well, this is where I'm split.
Mystery has always been an essential part of the human experience. Not knowing everything about the present and future has allowed for creativity to flourish and exploration thrive. But, if said mystery is reduced to a margin of triviality, then maybe the exploratory drive that has propelled humanity through the ages will be replaced by a sequential system with the sole goal of profit, eliminating many creative freedoms we now enjoy today. But, on the other hand, maybe they won't be. Maybe computer programs will never be able to map the impetuousness of human nature, and will simply allow us to explore more efficiently, with a more defined direction and with greater success.
OK, so the first thing we did in class today was observe an obese sumo wrestler plunging to his death on a pair
of skis he probably uses as tooth picks.
This was an intro, in some way or another, into a new topic that we will
be studying for the rest of the quarter: Related Rates.I will get to exactly what those are in a bit, but first a
quick review.
(fill in the blanks)
Calculus is the mathematics of _______, and everything changes with respect to ______.
Algebra is the study of the relationship between _______.
Calculus is the study of the relationship between the _______ __ _______ of the variables (echem… derivative)
Related Rates:
Related rates are, (as far as we are concerned), how multiple derivatives are related. These variable derivatives include...
dx/dt= instantaneous rate of change of horizontal displacement with respect to time (or as we will be looking at it for the rest of this scribe post, horizontal velocity )
dy/dt= instantaneous rate of change ofvertical displacement with respect to time (or as we will be looking at it for the rest of this scribe post, vertical velocity)
Take, for instance, a cone... (just bear with me, I did this all in Paint)
Say you were to fill that cone with some agua...
It becomes evident that the rate at which the height (vertical) of the water increases is directly dependent on the width (horizontal) of the cone. Both the rate of vertical growth and horizontal growth are unique derivatives, and they clearly have a relationship. (Hence the term related rates).
(answers to quiz): Change, time, variables, rate of change.
Next, we took a rubber pig, Porky, and ruthlessly pinned him against a wall using a long pvc pipe, mercilessly using him as a demonstration for our own convenience, completely oblivious to his own mutilated feelings. The situation is simply this: Porky is on top of the pipe, pinned against the wall. We decide to lower Porky by slowly sliding the bottom of the pipe away from the wall at a constant rate until Porky reaches the ground only to go into years of shock therapy.
The rest of class was dedicated to filling out Exploration 4.9A, which Bru so graciously posted.
The same scenario of Porky's descent is posed in the exploration. (In the exploration, the pipe becomes a ladder and the wall becomes a skyscraper). And the million dollar question that we answered in the investigation is...
How is the rate at which Porky is descending related to the rate at which the bottom of the ladder is moving way from the skyscraper?
In order to answer this, we need to find the velocity at which the bottom of the ladder moves away from the skyscraper and the rate at which Porky descends the building face.
The horizontal velocity is given to be 2 ft/s, the initial horizontal distance from the skyscraper is 10 ft, and the length of the ladder is 26 ft.
In order to find the vertical velocity, we need to find the height of porky at one second intervals. We can do this because we know that the bottom leg is expanding by two feet every second, and the hypotenuse stays the same.
If you are having trouble imagining this, here is a crude visual to help you out. Each colored triangle represents Porky and the ladder's position at different seconds.
Now comes the subject responsible for the largest amount of point deductions on our tests...Algebra!!!
We must use the Pythagorean theorem to determine the length of each part of the triangle at each second.
For instance...
if in the first second, the horizontal length is 10 ft, and the length of the ladder is 26 ft (see image above), we can use Pythag to determine the height of Porky.
a=10 ft
b=?
c=26 ft
b=24ft
Now, let's take the first second after the ladder is moved.
Because the ladder is moving at a constant rate of 2 ft/s away from the building, the initial length is 10 ft, and this is the first second, the horizontal length of the ladder from the building will be 12 ft from the skyscraper.
And since we know that the length of the ladder itself is constant, we can substitute the values
a=12 ft
b=?
c=26 ft
into the Pythagorean theorem, which gives us a b value of 23.06 ft.
This train of thought can be applied to the triangle every second until Porky reaches the ground, keeping in mind that the ladder's distance from the skyscraper will increase by two feet every second.
(Image taken from 3rd slide of 2-17)
After calculating all of the a and b distance values we are able to answer the initial question:
The rate at which Porky descends is related to the rate of the ladder's movement by some variable rate.
After we found all of the b values manually, we were asked to find an equation that related the changing a and b values.
For the sake of easiness, a and b can be substituted for x and y, respectively, leaving us with the equation...
To solve the unknown rate of Porky's velocity, we simply implicitly differentiate with respect to time...
This equation relates Porky's velocity and the ladder's velocity. It is called a Differential Equation because it contains a derivative.
So, that concludes Monday's lesson, the HW for Friday is pages 1-3 of the handout Bru gave us in class, Section 4-9A: Related Rates Warm Up.
PS, I kinda lied about being over with learning derivatives, I just really wanted to put a dancing cat in the post.
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