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4 problems match “dihedral angle”. Clear

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1. Volume of tetrahedron from dihedral angle ●●○○○ 20 ptsFaces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $30^\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.
2. Wedge cut from a cylindrical log ●●●○○ 50 ptsA cylindrical log has diameter $12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $45^\circ$ angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as $n\pi$, where $n$ is a positive integer. Find $n$.
3. Side length of a square tilted to a plane ●●●●○ 100 ptsGiven a square $ABCD$ with side length $a$, vertex $A$ lies in plane $\beta$, and the other vertices are on the same side of plane $\beta$. The distances from points $B$ and $D$ to plane $\beta$ are 1 and 2, respectively. If the angle between plane $ABCD$ and plane $\beta$ is $30^\circ$, find $a$. Round to two decimal places.
4. Angle between faces of an equifacial tetrahedron ●●●●○ 100 ptsLet $ABCD$ be a tetrahedron in which opposite edges are equal. The angle between planes $ABD$ and $BCD$ is $90^\circ$, and the angle between planes $BCD$ and $CAD$ is $60^\circ$. Find the angle between planes $CAD$ and $ABD$ in degrees.