Two spheres $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 81 , x ^ { 2 } + ( y - 5 ) ^ { 2 } + z ^ { 2 } = 56$ are denoted by $S _ { 1 } , S _ { 2 }$ respectively. Let P be a point on the circle formed by the intersection of the two spheres $S _ { 1 } , S _ { 2 }$, and let $\mathrm { P } ^ { \prime }$ be the orthogonal projection of point P onto the $xy$-plane. Let Q and R be the points where the sphere $S _ { 1 }$ intersects the $y$-axis. Find the maximum volume of the tetrahedron $\mathrm { PQP } ^ { \prime } \mathrm { R }$. [4 points]
In coordinate space, consider the triangle ABC with vertices $\mathrm { A } ( 54,0,0 ) , \mathrm { B } ( 0,27,0 ) , \mathrm { C } ( 0,0,27 )$ on the plane $x + 2 y + 2 z = 54$. A point $\mathrm { P } ( x , y , z )$ is in the interior of triangle ABC. Let Q be the orthogonal projection of P onto the $xy$-plane, R be the orthogonal projection of P onto the $yz$-plane, and S be the orthogonal projection of P onto the $zx$-plane. When $\overline { \mathrm { QR } } = \overline { \mathrm { QS } }$, find the maximum volume of the tetrahedron QPRS. [4 points]
In coordinate space, one face ABC of a regular tetrahedron ABCD lies on the plane $2 x - y + z = 4$, and the vertex D lies on the plane $x + y + z = 3$. When the centroid of triangle ABC has coordinates $( 1,1,3 )$, what is the length of one edge of the regular tetrahedron ABCD? [4 points] (1) $2 \sqrt { 2 }$ (2) 3 (3) $2 \sqrt { 3 }$ (4) 4 (5) $3 \sqrt { 2 }$
In coordinate space, there is a sphere $$S : ( x - 2 ) ^ { 2 } + ( y - \sqrt { 5 } ) ^ { 2 } + ( z - 5 ) ^ { 2 } = 25$$ with center $\mathrm { C } ( 2 , \sqrt { 5 } , 5 )$ passing through point $\mathrm { P } ( 0,0,1 )$. For a point Q moving on the circle formed by the intersection of sphere $S$ and plane OPC, and a point R moving on sphere $S$, let $\mathrm { Q } _ { 1 }$ and $\mathrm { R } _ { 1 }$ be the orthogonal projections of points $\mathrm { Q }$ and $\mathrm { R }$ onto the xy-plane respectively. For two points $\mathrm { Q } , \mathrm { R }$ that maximize the area of triangle $\mathrm { OQ } _ { 1 } \mathrm { R } _ { 1 }$, the area of the orthogonal projection of triangle $\mathrm { OQ } _ { 1 } \mathrm { R } _ { 1 }$ onto plane PQR is $\frac { q } { p } \sqrt { 6 }$. Find the value of $p + q$. [4 points]
In rectangular parallelepiped $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, $A B = B C = 2$, and the angle between $A C _ { 1 }$ and plane $B B _ { 1 } C _ { 1 } C$ is $30 ^ { \circ }$. Then the volume of the rectangular parallelepiped is A. 8 B. $6 \sqrt { 2 }$ C. $8 \sqrt { 2 }$ D. $8 \sqrt { 3 }$
The points $A ( 6 | 0 | 4 ) , B ( 0 | 6 | 4 ) , C ( - 6 | 0 | 4 )$ and $D$ lie in the plane $E$ and form the vertices of the square base of a pyramid ABCDS with apex $S ( 0 | 0 | 1 )$. $A , B$ and $S$ lie in the plane $F$. Calculate the volume $V$ of the pyramid ABCDS.
Water fountains emerge at four points on the surface of the marble sphere. One of these exit points is described in the model by the point $L _ { 0 } ( 1 | 1 | 6 )$. The corresponding fountain is modeled by points $L _ { t } \left( t + 1 | t + 1 | 6,2 - 5 \cdot ( t - 0,2 ) ^ { 2 } \right)$ with suitable values $t \in \mathbb { R } _ { 0 } ^ { + }$. A total of 80 ml of water flows per second from the four exit points into the bronze bowl. Determine the time in seconds that passes until the initially empty fountain is completely filled with water.
A regular tetrahedron has all its vertices on a sphere of radius $R$. Then the length of each edge of the tetrahedron is (a) $( \sqrt{2} / \sqrt{3} ) R$ (b) $( \sqrt{3} / 2 ) R$ (c) $( 4 / 3 ) R$ (d) $( 2 \sqrt{2} / \sqrt{3} ) R$
The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $\hat { a } , \hat { b } , \hat { c }$ such that $$\hat { a } \cdot \hat { b } = \hat { b } \cdot \hat { c } = \hat { c } \cdot \hat { a } = \frac { 1 } { 2 }$$ Then, the volume of the parallelopiped is (A) $\frac { 1 } { \sqrt { 2 } }$ (B) $\frac { 1 } { 2 \sqrt { 2 } }$ (C) $\frac { \sqrt { 3 } } { 2 }$ (D) $\frac { 1 } { \sqrt { 3 } }$
In coordinate space, consider the circle of radius $1$ centered at the origin in the $xy$-plane. Let $S$ be the cone (including its interior) with this circle as its base and with vertex at the point $(0,\,0,\,2)$. Also, let $A(1,\,0,\,2)$.
[(1)] When point $P$ moves over the base of $S$, let $T$ be the region swept out by the line segment $AP$. Illustrate in the same plane the cross-section of $S$ by the plane $z=1$ and the cross-section of $T$ by the plane $z=1$.
[(2)] When point $P$ moves throughout $S$, find the volume of the region swept out by the line segment $AP$.