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| Geometry - solid figures
solved problems |
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Solid figures, solved problems examples |
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| Example:
An equilateral triangle of the side a =
6 rotates around a line parallel to its side on the distance
that equals the
triangle’s height (not passing through the vertex of the triangle), find the volume of the solid of revolution. |
| Solution: Given a =
6. Vsolid
of revolution = ?
Vsolid
of revolution =
2Vfrustum
-
Vcylinder |
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| Example:
The cosine of the angle at which diagonals of a cube intersect is? |
| Solution: Given
a cube.
cos j
= ? |
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| Example:
In a right triangular prism inscribed is a sphere which touches all faces of the prism. The ratio of surfaces of the
sphere and the prism is? |
| Solution: Given
a right triangular prism.
Ssphere
: Sprism
= ?
Ssphere
= 4R2p, |
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| Example:
In a cube inscribed is a regular tetrahedron whose edges are face’s diagonals of the cube, the ratio of volumes of
the cube and the tetrahedron is? |
| Solution: Given
a cube.
Vcube
: Vtetrahedron
= ?
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| Example:
At what distance from the center of a not-transparent sphere of radius
4
cm
should be placed
the light to brighten
up one third of the surface of the sphere. |
| Solution: Given
r =
4
cm. SO
= ? |
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| Example:
In an oblique circular cone the longest slant height
s1=
6Ö3
and the shortest s2=
6
subtend, at the vertex of the
cone, the angle of 30°, find the volume of the cone. |
| Solution: Given
s1=
6Ö3
and s2=
6. Vcone = ? |
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| Example:
A plane passes through three vertices of a cube and cuts of it a tetrahedron.
The ratio of the volumes of obtained
solids is? |
| Solution: Given a cube.
Vtetrahedron
:
(Vcube
- Vtetrahedron
) = ?
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| Example:
Around a cone of slant height that equal the diameter of the base circumscribed is a sphere.
The
ratio of surfaces of the cone
and the sphere is? |
| Solution: Given
s = 2r.
Scone
: S
sphere
= ? |
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| Example:
A point source of light is 4
m distant from the center of a sphere of radius
2
m. What surface area of the sphere
is brightened? |
| Solution: Given
R
= 2 m and
SA
= 4 m. S
cap
= ? |
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| Geometry
and use of trigonometry contents - B |
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