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Mathematics-Online problems: | ||
Solution to the problem of the (previous) week |
Problem:
Find the points of intersection
on the edges.
Compute the volumes of the tetrahedron and of the solid subfigure containing the origin .
Hint:
The dashed triangle
dissects one of the solid subfigures into a prism and a pyramid having a quadrilateral base.
Answer:
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( , , ) | ||
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( , , ) | ||
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( , , ) | ||
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( , , ) | ||
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(The results should be correct to three decimal places.)
Solution:
We obtain the points of intersection on the edges by inserting the parametric representation of the edge into the equation of the plane.
i. e.
i. e.
i. e.
and
i. e.
and
The area of the base triangle of the tetrahedron is
Hence, for the volume of the tetrahedron we get
The prism with base
and height
has the volume
The base of the pyramid is a trapezoid with vertices
.
Its area is
Hence, the solid subfigure containing the origin has the volume