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Mathematics-Online problems:

# Solution to the problem of the (previous) week

Problem:

#./interaufg69_en.tex#The plane

dissects the tetrahedron with vertices

into two solid subfigures.

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.

 ( , , ) ( , , ) ( , , ) ( , , )

(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

with being the vertex of the tetrahedron.

The prism with base and height has the volume

The base of the pyramid is a trapezoid with vertices .
Its area is

Multiplying the area with one third of the height , we obtain the volume

Hence, the solid subfigure containing the origin has the volume

[problem of the week]