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ANIMATIONS FOR ALGEBRA AND PRECALCULUS
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Click on the ellipse and hyperbola at
the left to see
animations demonstrating the foci definitions of each.
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You should be able to compute the total distance traveled by an idealized
bouncing ball dropped from a given initial height that always bounces back up to
some fraction (less than one) of the height it just fell from. The first
five people to correctly identify the fraction (ratio r) in this bouncing
ball animation will receive a bonus.
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| A red train leaves Altoona heading to
Spoker at 4PM at a speed of 65mph. One hour later a blue train leaves Spoker heading to
Altoona at a speed of 80mph. The distance from Altoona to Spoker is
355 miles. At what time will the trains pass each other? Which
train will arrive at its destination first? Click
here to see an animation. In the animation the path of the red
train is in red and the time the red train has been traveling is shown vertically in
red. The path of the blue train is in blue and the time the blue
train has been
traveling is shown vertically in blue. Click
here for another version of the animation where little trains are
drawn instead of colored paths. Quicktime
animation |
| Zeno's Paradox
The Greek philosopher Zeno (495 - 435 B.C.) may
have been a fast runner at one time and perhaps claimed that if he was
given a slight head start no one on earth could catch him. If so
then by the paradox named after him he would still have made the claim
even after he aged and was no longer a fast runner. This is because,
as he said, to catch him one must get to where he was and by then he will
have moved forward. You can never get to where he is because to get
to where he is you must get to where he was first and by the time you get
to where he was he will no longer be there. This implies that to
score a touchdown in American football a wide receiver has merely to catch
the ball behind the secondary. No matter how fast the defensive
backs are they will not be able to catch up to the receiver and the
receiver can simply walk to the end zone, taking care never to pause along
the way. As long as he never stops moving and continues heading
toward the end zone no one can ever reach his position from behind
him. He would have to stop moving away from the quarterback briefly
to catch the football since otherwise the football could never catch up to
him.
To present this paradox mathematically, Zeno
considered the case of Achilles and the tortoise. I will consider an
aged Achilles and the tortoise although the same argument would apply to a
young fleet Achilles and the tortoise. Suppose Achilles runs at a
rate of 1 meter/sec and the tortoise crawls at a rate of 9/10
meter/sec. If the tortoise is given a 1 meter head start and
continues to travel at a rate of 9/10 m/sec then Achilles traveling at 1
m/sec cannot catch up to him. This is because once Achilles has
traveled the 1 meter to get to where the tortoise was when the chase
began, the tortoise will have traveled 9/10 of a meter farther. Once
Achilles travels that 9/10 of a meter the tortoise will have traveled an
additional (9/10)2 meter or 81/100 meter. Once Achilles
travels this 81/100 meter the tortoise will have traveled an additional
(9/10)3 meter or 729/1000 meter, and so on so Achilles will
never catch the tortoise. Achilles will travel a distance of
and in spite of n approaching infinity he will
never catch the tortoise. Can you mathematically demonstrate the
flaw in this argument?
In the animation
of this paradox the tortoise is in blue, Achilles in red, and the time
that has elapsed during the chase is shown in green along the y-axis.
Quicktime animation |
| A jet is descending toward the ground
along a path described by the upper branch of the hyperbola 16y2
- x2 = 400. When the jet reaches the vertex of this
hyperbolic path it will begin gaining altitude again along the same
hyperbolic path. The y-coordinate gives the jet's height above the
ground in meters. How close to the ground does the jet come?
Along the path of the jet, 100 meters beyond its lowest point, there is a
building 25 meters high. Will the jet clear the building? If
yes, by how much, and if no, by how much? Give your answer to the
nearest 1/1000 of a meter. Click on the picture at the right to see
an animation and click here to see an
animation with scales. |
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| A bird is perched at the top of a
pole that is 10 feet high. It flies in an elliptical path to the top
of a pole that is 30 feet high and 50 feet away from the 10 foot
pole. The center of the elliptical path is directly above the 10
feet pole and at the same height as the 30 foot pole. It takes the
bird 10 seconds to fly from the top of the 10 foot pole to the top of the
30 foot pole. Give a parametric representation of the path of the
bird where t = 0 represents the time when the bird leaves the 10 foot
pole. Click here to see an
animation of the bird's flight from pole to pole. |
| Finding a polynomial function
Your task is to find the fourth degree
polynomial function that gives the path followed by the tip of the beak of
the bird in the animation. In the animation
the coordinates of the end of the orange branch at the far left are
(1,27). The coordinates of the end of the green branch are (10,20),
the coordinates of the end of the red branch are (18,31), and the
coordinates of the end of the purple branch are (21,28). The
coordinates of the end of the orange branch at the far right are
(26,39). Distance is in feet. The tip of the beak of the bird
is at (1,27), (the end of the orange branch at the left), when the bird
falls off the branch. The bird begins to fly and its path takes the
tip of its beak through points exactly 2 feet above the tips of the green,
red, purple, and orange (on the right) branches. It is autumn.
Click here to see the animation
without scales on the x- and y-axis. |
| This animation is meant to
demonstrate the notion that the area of a triangle is one half the area of
a rectangle whose length is equal to the length of one of the bases of the
triangle and whose width is equal to the triangle's altitude to that
base. The notion is established using similar triangles. Click
on any one of the pictures on the right to see an animation of the
triangle moving into the rectangle along a semi-circular path. |
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| Two men leave the same place at the
same time walking in opposite directions. One man (Blue) is walking
at the rate of 2 mph and the other man (Green) is walking at the rate of 3
mph. How long before they will be 6 miles apart? How far apart
are they after 2 hours? Click here
to see an animation. In the animation the elapsed time is shown by
the animated point on the y-axis. The x-axis indicates distance
traveled corresponding to the two animated men. The distance
traveled by Blue is indicated by negative numbers since his direction is
opposite that of Green. Quicktime
animation |
| Imagine a teeter totter (red) that is
4 meters long and sqrt(2) meters above the ground at its center (where it
is supported--green support). Initially the teeter totter is at a
angle of 45o below the horizontal (in blue). The motion
of the teeter totter is such that on its first rotation upward it goes
through an angle of (17/18)(90) = 85o. On its subsequent
rotation downward it passes through an angle of (17/18)(17/18)(90)
degrees. On its next rotation upward it passes through an angle of
(17/18)(17/18)(17/18)(90) degrees. This pattern continues
indefinitely. What is the limit to the distance traveled by a point
on the extreme tip of the teeter totter? Click here to see an
animation. The animation goes through a few rotations of the teeter
totter. It is not designed to be completely accurate but should give
you a rough idea of what is happening. |
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