The theorem that Archimedes was the most proud of (so much that he requested it to be engraved on his tomb) was that the volume of a sphere is exactly of the volume of the cylinder circumscribing it. The goal of this exercise is to demonstrate this with the method of mechanical theorems. To simplify the calculations, we will work with the usual coordinate system. Consider the unit circle centered at (0,1), and the lines x = 1 and y = x. We respectively get a half-sphere, a cylinder and a cone (all of height 1) by revolving those curves about the y-axis, for y € [0, 1] (see picture). Y B A B C/ D Η • (a) Let y = [0, 1]. Draw a horizontal line through the point A = (0, y), and let B, C, D be the points, with increasing a values, where that line intersects the three curves (see picture). Show that the disk that we get by revolving the segment AD about the y-axis is in equilibrium with the sum of the two disks, with radius AB and AC, when their centers are moved to the point H = (0, –¹). Here, the lever is the y-axis, and the fulcrum is at the origin. (b) Archimedes would conclude that the cylinder where it is is in equilibrium with the sum of the half-sphere and the cone, when their centers of gravity are both at H. Use this and the fact (known to Archimedes) that the volume of a cone inscribed in a cylinder is exactly of that cylinder, to deduce that the volume of the sphere in this problem is of the volume of the cylinder of height 2 circumscribing it.
The theorem that Archimedes was the most proud of (so much that he requested it to be engraved on his tomb) was that the volume of a sphere is exactly of the volume of the cylinder circumscribing it. The goal of this exercise is to demonstrate this with the method of mechanical theorems. To simplify the calculations, we will work with the usual coordinate system. Consider the unit circle centered at (0,1), and the lines x = 1 and y = x. We respectively get a half-sphere, a cylinder and a cone (all of height 1) by revolving those curves about the y-axis, for y € [0, 1] (see picture). Y B A B C/ D Η • (a) Let y = [0, 1]. Draw a horizontal line through the point A = (0, y), and let B, C, D be the points, with increasing a values, where that line intersects the three curves (see picture). Show that the disk that we get by revolving the segment AD about the y-axis is in equilibrium with the sum of the two disks, with radius AB and AC, when their centers are moved to the point H = (0, –¹). Here, the lever is the y-axis, and the fulcrum is at the origin. (b) Archimedes would conclude that the cylinder where it is is in equilibrium with the sum of the half-sphere and the cone, when their centers of gravity are both at H. Use this and the fact (known to Archimedes) that the volume of a cone inscribed in a cylinder is exactly of that cylinder, to deduce that the volume of the sphere in this problem is of the volume of the cylinder of height 2 circumscribing it.
Elementary Geometry for College Students
6th Edition
ISBN:9781285195698
Author:Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:Daniel C. Alexander, Geralyn M. Koeberlein
Chapter9: Surfaces And Solids
Section9.3: Cylinders And Cones
Problem 48E
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![The theorem that Archimedes was the most proud of (so much that he requested it to
be engraved on his tomb) was that the volume of a sphere is exactly 3 of the volume of
the cylinder circumscribing it. The goal of this exercise is to demonstrate this with the
method of mechanical theorems.
To simplify the calculations, we will work with the usual coordinate system. Consider
the unit circle centered at (0, 1), and the lines x = 1 and y = x. We respectively get a
half-sphere, a cylinder and a cone (all of height 1) by revolving those curves about the
y-axis, for y = [0, 1] (see picture).
S
A
н.
B C D
X
(a) Let y = [0, 1]. Draw a horizontal line through the point A = (0, y), and let B, C, D
be the points, with increasing x values, where that line intersects the three curves
(see picture). Show that the disk that we get by revolving the segment AD about
the y-axis is in equilibrium with the sum of the two disks, with radius AB and AC,
when their centers are moved to the point H = (0, -1). Here, the lever is the y-axis,
and the fulcrum is at the origin.
(b) Archimedes would conclude that the cylinder where it is is in equilibrium with the
sum of the half-sphere and the cone, when their centers of gravity are both at H.
Use this and the fact (known to Archimedes) that the volume of a cone inscribed
in a cylinder is exactly of that cylinder, to deduce that the volume of the sphere
in this problem is of the volume of the cylinder of height 2 circumscribing it.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa039dd32-ee56-4bfe-8a77-81df3f075482%2F0fca169c-a576-4e3a-b7e5-fe6d5edb1433%2Fimetzr_processed.png&w=3840&q=75)
Transcribed Image Text:The theorem that Archimedes was the most proud of (so much that he requested it to
be engraved on his tomb) was that the volume of a sphere is exactly 3 of the volume of
the cylinder circumscribing it. The goal of this exercise is to demonstrate this with the
method of mechanical theorems.
To simplify the calculations, we will work with the usual coordinate system. Consider
the unit circle centered at (0, 1), and the lines x = 1 and y = x. We respectively get a
half-sphere, a cylinder and a cone (all of height 1) by revolving those curves about the
y-axis, for y = [0, 1] (see picture).
S
A
н.
B C D
X
(a) Let y = [0, 1]. Draw a horizontal line through the point A = (0, y), and let B, C, D
be the points, with increasing x values, where that line intersects the three curves
(see picture). Show that the disk that we get by revolving the segment AD about
the y-axis is in equilibrium with the sum of the two disks, with radius AB and AC,
when their centers are moved to the point H = (0, -1). Here, the lever is the y-axis,
and the fulcrum is at the origin.
(b) Archimedes would conclude that the cylinder where it is is in equilibrium with the
sum of the half-sphere and the cone, when their centers of gravity are both at H.
Use this and the fact (known to Archimedes) that the volume of a cone inscribed
in a cylinder is exactly of that cylinder, to deduce that the volume of the sphere
in this problem is of the volume of the cylinder of height 2 circumscribing it.
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