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Angles in pyramids

Solve the first problem 15 problems

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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. Angle CBD in a tetrahedron ●●○○○ 20 ptsIn tetrahedron $ABCD$, $BC=1$, $BD=2$, $\angle ABC=90^{\circ}$, and $\overrightarrow{BC}\cdot\overrightarrow{DA}=-\sqrt{3}$. Find $\angle CBD$ in degrees.
3. Area ratio of faces in a pyramid ●●●○○ 50 ptsThe height of the triangular pyramid $ABCD$, dropped from vertex $D$, passes through the point of intersection of the heights of triangle $ABC$. In addition, it is known that $DB=3$, $DC=2$, $\angle BDC=90^{\circ}$. Find the ratio of the area of face $ADB$ to the area of face $ADC$.
4. Area of triangle BDE in space ●●●○○ 50 ptsPoints $A$, $B$, $C$, $D$, and $E$ are located in 3-dimensional space with $AB= BC= CD= DE= EA= 2$ and $\angle ABC = \angle CDE = \angle DEA = 90^\circ$. The plane of triangle $ABC$ is parallel to $\overline{DE}$. What is the area of triangle $BDE$?
5. Dot product in a regular tetrahedron ●●●○○ 50 ptsGiven a regular tetrahedron $P-ABC$ with edge length $1$, points $E, F, G, H$ lie on edges $PA, PB, CA, CB$ respectively. If $\overrightarrow{EF}+\overrightarrow{GH}=\overrightarrow{AB}$ and $\overrightarrow{EH} \cdot \overrightarrow{FG}=\frac{1}{18}$, find $\overrightarrow{EG} \cdot \overrightarrow{FH}$. Round to two decimal places.
6. Sum of squared tangents in a tetrahedron section ●●●●○ 100 ptsA regular tetrahedron is intersected by a plane that is perpendicular to one of its faces and meets that face along a line $e$. The intersection lines of this plane with the other three face planes of the tetrahedron form angles $\varphi_1, \varphi_2, \varphi_3$ with $e$. Determine the value of $\tan^2 \varphi_1+\tan^2 \varphi_2+\tan^2 \varphi_3$.
7. 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.
8. 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.

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