As we continue the “Why Study Math” series of articles, here we look at the conic section called the hyperbola. The hyperbola is obtained by intersecting the double-napped cone (see the other articles in this series on this point) with a plane so that both parts of the cone are cut. Those familiar with the parabola might note that this curve almost looks like two parabolas pasted back to back with a space in between them. Mathematically, the hyperbola is not a parabola, although these two conic sections have a similar outward appearance.
The hyperbola is the least known of the four conic sections. It is also the most difficult curve to derive algebraically. Probably for this reason, students who study the conic sections, like the hyperbola the least. However, when students see the reason we study this curve, their attitude changes significantly. For this reason, we will now examine some of those applications connected to the hyperbola.
Everyone at one time or another has thrown a pebble into a still pond. Picture throwing not one but two pebbles into this pond at the same time. The outward concentric circles that form intersect each other at points which trace out the curve known as the hyperbola. This application is used in radar tracking stations. LORAN, the terrestrial navigation system, uses low frequency radio transmitters to locate objects. Objects are located by sending out sound signals from two sources to a receiving station, such as one found on a boat or plane. The constant time difference between the signals from the two stations is represented by a hyperbola.
As we discussed with the applications of the ellipse, most celestial bodies follow elliptical orbits. In the case of comets, however, a hyperbolic path is followed as they shoot through space. The hyperbola is also the shadow cast on a wall by a lamp with a cylindrical shade. And for something a little more earthy, the shape of that horse saddle you get on to ride forms an interesting solid curve called a hyperbolic paraboloid. So you see, the conic sections–even the hyperbola–might be closer than you think.
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By: Joe Pagano
Why Study Math? – The Hyperbola
by AdminFinding a Mall Parking Spot Using Mathematics – Part I
by Admin
I am always amused when people become fascinated with simple demonstrations of the power of even basic mathematics. We have become mostly a jaded society depending on the technological progress of corporations to give us more and more electronic gadgetry and the like. We have forgotten how all this technological stuff originates: via the harnessing of the knowledge of mathematics and science and their applications in practical ways.
Thus as educator and mathematician, I always enjoy a big smile when I can demonstrate the practicality of knowing even rudimentary mathematics. A good and solid understanding of basic geometry, arithmetic, and algebra can go a long way toward understanding many fundamental laws of nature and even permitting a high degree of general problem solving abilities. People are always amazed when I explain to them how I can tell time at the beach using the sun, or use probability to make general predictions, or to do seemingly amazing human calculator multiplications in my head. Yet all these feats are quite achievable even for the ordinary person.
Having said this, I can move toward the interesting sounding title of this article and explain how basic mathematics can be used for even such an odd sounding thing as finding a parking spot in a busy mall. We all know how frustrating this can be particularly around the holiday shopping season. Now man is a clever animal and if you ask people their method many will quickly volunteer that they have a great system. These systems range from stalking people who leave the mall to circling like hungry sharks waiting to feed on the next open spot.
Personally, I had always had a problem with the methods mentioned above and for this reason, I suppose, one day I thought about the problem and asked myself whether mathematics could solve this problem–or at least come up with a more practical method. Being a mall rat largely because of my wife’s intense love of such locale, I had many opportunities to drop her off in front and then begin the process of “search and destroy,” or more simply put, to wait for parking spots. Being a fan of the two mathematical disciplines of probability and statistics, I decided to see whether using some basic facts within these two areas could help me solve the problem. Thus sitting by the mall waiting for an open spot (mind you the times we would go to the mall were usually on Saturdays when choice spots were rare, and I would always want to park in a particular area near Macy’s where spots were even more limited because of the smaller parking area), I began to make some basic assumptions and to cogitate deeply.
Almost as by happenstance, I pieced together a rudimentary method, did some quick calculations and tested the underlying hypotheses. I surveyed the fully occupied parking area, targeted my preferred area to park, and predicted that within an interval of no more than five minutes one of those preferred spots would become available. Lo and behold within five minutes, a spot opened up. I was amused. Over the next ten or fifteen visits to the mall I tried and tested this method. Success after success after success. I analyzed both the basic assumptions and mathematics used and tested again and again. Always worked. I even demonstrated the method with people in the car. The reaction was always one of amazement when I could tell them within how long a spot would open up. To them, this always seemed like magic; however, a little thoughtful contemplation joined to some basic mathematics was the glue holding the mix together.
In a follow-up article, I will reveal the method and the basic underlying assumptions, as well as the mathematical principles involved (for those who are squeamish about mathematics, fear not; for the explanations will not involve anything beyond the scope of layman mathematics, and my particular strength is the ability to boil down the esoteric and make it understandable). For those who have read this far, I think you realize the broader implications of this: if mathematics can help you find a parking spot in a busy mall, what else can it do? See you in Part II.
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By: Joe Pagano
