Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However. perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the "versoria," a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, "versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a circle of radius a so that the points ( 0 , 0 ) and ( 0 , 2a ) are points on the circle (Figure 1.12). Let O denote the origin. Choose any other point A on the (tilde, and draw the secant line OA . Let B denote the point at which the line OA intersects the horizontal line through ( 0 , 2a ) . The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution, In this project you will parameterize these curves. Figure 1.12 As the point A moves around the circle, the point P traces out the witch of Agnesi curve for the given circle. 3. Note that x = d cos θ . Show that x = 2 a cot θ . When you do this, you will have parameterized the x -coordinate of the curve with respect to θ . If you can get a similar equation for y , you will have parameterized the curve.
Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However. perhaps the strangest name for a curve is the witch of Agnesi. Why a witch? Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the "versoria," a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, "versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since. The witch of Agnesi is a curve defined as follows: Start with a circle of radius a so that the points ( 0 , 0 ) and ( 0 , 2a ) are points on the circle (Figure 1.12). Let O denote the origin. Choose any other point A on the (tilde, and draw the secant line OA . Let B denote the point at which the line OA intersects the horizontal line through ( 0 , 2a ) . The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle. Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution, In this project you will parameterize these curves. Figure 1.12 As the point A moves around the circle, the point P traces out the witch of Agnesi curve for the given circle. 3. Note that x = d cos θ . Show that x = 2 a cot θ . When you do this, you will have parameterized the x -coordinate of the curve with respect to θ . If you can get a similar equation for y , you will have parameterized the curve.
Many plane curves in mathematics are named after the people who first investigated them, like the folium of Descartes or the spiral of Archimedes. However. perhaps the strangest name for a curve is the witch of Agnesi. Why a witch?
Maria Gaetana Agnesi (1718—1799) was one of the few recognized women mathematicians of eighteenth-century Italy. She wrote a popular book on analytic geometry, published in 1748, which included an interesting curve that had been studied by Fermat in 1630. The mathematician Guido Grandi showed in 1703 how to construct this curve, which he later called the "versoria," a Latin term for a rope used in sailing. Agnesi used the Italian term for this rope, "versiera,” but in Latin, this same word means a "female goblin.” When Agnesi’s book was translated into English in 1801, the translator used the term “witch” for the curve, instead of rope. The name “witch of Agnesi” has stuck ever since.
The witch of Agnesi is a curve defined as follows: Start with a circle of radius a so that the points
(
0
,
0
)
and
(
0
,
2a
)
are points on the circle (Figure 1.12). Let O denote the origin. Choose any other point A on the (tilde, and draw the secant line OA. Let B denote the point at which the line OA intersects the horizontal line through
(
0
,
2a
)
. The vertical line through B intersects the horizontal line through A at the point P. As the point A varies, the path that the point P travels is the witch of Agnesi curve for the given circle.
Witch of Agnesi curves have applications in physics, including modeling water waves and distributions of spectral lines. In probability theory, the curve describes the probability density function of the Cauchy distribution, In this project you will parameterize these curves.
Figure 1.12 As the point A moves around the circle, the point P traces out the witch of
Agnesi curve for the given circle.
3. Note that
x
=
d
cos
θ
. Show that
x
=
2
a
cot
θ
. When you do this, you will have parameterized the x-coordinate of the curve with respect to
θ
. If you can get a similar equation for y, you will have parameterized the curve.
In areas using rose-petal curves in the form of r=acos(theta) with the same length, does the equation with greater number of petals always mean a larger area?
Please illustrate and explain why.
The points A, B, C and D lie on the circle shown in the figure such that AB = BC and DC // AB. The tangent drawn to the circle at A is AL.
Per.:
HIGH DIVE Activity
The following problem is not the same as our HIGH DIVE PROBLEM, it merely allows you to practice
calculating heights and distances in correlation with our problem. The dimensions of the Ferris Wheel
in this problem are different than our High Dive problem! Add your work
The Ferris Wheel
Al and Betty have gone to the amusement park to ride on a Ferris wheel. The wheel in the park has a
radius of 15 feet, and its center is 20 feet above ground level. Assume it takes 24 seconds to make a
complete revolution.
Name:
You can describe the various positions in the cycle of the Ferris wheel in terms of the face of a clock, as
indicated in the accompanying diagram. Think of Al and Betty's location as they ride as simply a point
on the circumference of the wheel's circular path. That is, ignore the size of the Ferris wheel seats, Al
and Betty's own heights, and so on.
ANSWER THE FOLLOWING QUESTIONS.
Complete any necessary work on a
separate sheet of paper and attach to…
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