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Mathematics-Online problems:

Problem of the week

The figure shows a tetrahedron spanned by the points

$\displaystyle (0,0,0)\,,\quad (1,0,0)\,,\quad (0,1,0)\,,\quad (0,0,1)\,, $

and intersected by the plane

$\displaystyle x+2y+2z=s\ . $

\includegraphics[width=0.45\linewidth]{tetraeder1.eps}         \includegraphics[width=0.45\linewidth]{tetraeder2.eps}

For an arbitrary constant $ s\in\mathbb{R}$, determine the points, where the plane and the edges of the tetrahedron intersect. Moreover, determine the areas of the resultant trapezoids $ (s\in(1,2))$ and triangles $ (s\in(0,1])$.

Answer:

Trapezoids
$ A$ $ \left(\right.$$ s +$ $ \mid $ $ s +$ $ \mid $ $ s +$ $ \left.\right)$
$ B$ $ \left(\right.$$ s +$ $ \mid $ $ s +$ $ \mid $ $ s +$ $ \left.\right)$
$ C$ $ \left(\right.$$ s +$ $ \mid $ $ s +$ $ \mid $ $ s +$ $ \left.\right)$
$ D$ $ \left(\right.$$ s +$ $ \mid $ $ s +$ $ \mid $ $ s +$ $ \left.\right)$
Area $ s^2 +$$ s +$


Triangles
$ A$ $ \left(\right.$$ s +$ $ \mid $ $ s +$ $ \mid $ $ s +$ $ \left.\right)$
$ B$ $ \left(\right.$$ s +$ $ \mid $ $ s +$ $ \mid $ $ s +$ $ \left.\right)$
$ C$ $ \left(\right.$$ s +$ $ \mid $ $ s +$ $ \mid $ $ s +$ $ \left.\right)$
Area $ s^2 +$ $ s +$


   

[solution to the problem of the previous week]