| Chapter 11: Additional
Topics in Analytic Geometry |
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Exercise 1:
Go to Newtonian
Gravitation and the Laws of Kepler and read about how conic sections are demonstrated
in the motions of the heavenly bodies (planets, comets, etc.). Find the following:
| 1) |
Two examples of bodies with almost
circular orbits (stating one focus of each) |
| 2) |
Two examples of bodies with
elliptical orbits (stating one focus of each) |
| 3) |
One examples of a body with a
parabolic orbit |
| 4) |
One example of a body with
hyperbolic motion |
| For the last two, you
may need to investigate this site further. (Try going home to see what is there.) |
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Exercise 2:
Parabolas aren't just very important to the
study of astronomy because they describe the paths of some comets! Did you see the movie
Contact? Visit The Circle of
Curvature, the Big Dish, and the Little Green Men and give a brief mathematical
description of big dish at Arecibo and write a paragraph about why they are using the
dish. Now go read about Liquid
Mirror Telescopes. Write a paragraph on the advantages of this technology and how it
works. |
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Exercise 3:
Exercises 1 and 2 had you investigate how
parabolas are used in space exploration. To see a demonstration of how the parabolic
dishes collect their information, go try this Focus of a Parabola applet. Do
you think a telescope would be more effective with the dish being wider or narrower? |
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Exercise 4:
This JAVA applet, Drawing Parabolas,
will allow you to change the values of a, the distance from the vertex to the focus, and
see what happens with the graph. Read the instructions and answer the question asked on
the site. |
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Exercise 5:
Visit this parabola JAVA applet and
experiment with the values of p. (Note that they are using the letter "p" where
our text uses an "a".)Let p = 50, then p = 25, then p = 2, then p = -2, then p = -25
and, finally p = -50.
What conclusions can you draw based
on what you've seen? What happens when you let p = 0? Now let p = 0 in your standard
equation for the parabola. What happened?
(answer)
********* NOTE!! CHECK a = p!!!!!! |
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Exercise 6:
Did you know that, if you were in a room with
an ellipsoidal ceiling and you were positioned at one focus with your friend at the other,
you could whisper across a crowed room and, not only would your friend here you perfectly,
no one else would be able to? To appreciate the power of the ellipse, visit Reflective
Properties of Ellipses. When you play with this applet, picture the whisper shooting
from one focus to the other. Also, be sure to move the two foci together to see what
happens. |
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Exercise 7:
Now that you've seen how particles (or
whispers) can "bounce around" an ellipse, go to Reflective Properties
of Hyperbolas and see what happens with the hyperbolic shape. |
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Exercise 8:
Go to Key Curriculum Press' site, The Conic Sections as the
locus of Perpendicular Bisectors. It will take a few minutes to load. Play with the
JAVA applet by dragging the red dots around with your mouse. Specifically, notice what is
happening with the eccentricity while you have an ellipse. Answer the site's first three
questions. |
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Exercise 9:
To investigate parametric curves a little more
quickly, let's visit this special parametric curve
calculator. When you first enter the site, these equations will appear:x = t * cos(t)
y = t * sin(t)
(Note: You will have to scroll down
to see the graph!)
Go ahead and plot these and make a
sketch (by hand) of the graph you see. Now, change the equations to:
x = 5* cos(t)
y = 5 * sin(t)
What happened to the graph? (Be
sure to pay close attention to the "tick" marks on the x and y axies!) Now try:
x = 5 * cos(t)
y = 15 * sin(t)
What happened this time? Arrive at
a conclusion about what was happening with the first set of parametric equations you
graphed as opposed to selecting specific values (like 5 and 15)... What caused the first
set to form the spiral?
(answer) |
