In coordinate space, the $xy$-plane, $yz$-plane, and $zx$-plane divide the space into 8 regions. How many of these 8 regions does the sphere
$$( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = 24$$
pass through? [4 points]
(1) 8
(2) 7
(3) 6
(4) 5
(5) 4

In coordinate space, a sphere $S$ with center coordinates all positive has center at $(x, y, z)$ where $x > 0, y > 0, z > 0$, is tangent to the $x$-axis and $y$-axis respectively, and intersects the $z$-axis at two distinct points. The area of the circle formed by the intersection of sphere $S$ and the $xy$-plane is $64 \pi$, and the distance between the two intersection points with the $z$-axis is 8. What is the radius of sphere $S$? [4 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15

In coordinate space, find the sum of all real numbers $k$ such that the plane $x + 8 y - 4 z + k = 0$ is tangent to the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 2 y - 3 = 0$. [3 points]

15. A quadrangular pyramid $P - A B C D$ has all vertices on the surface of sphere $O$. $PA$ is perpendicular to the plane containing rectangle $A B C D$. $AB = 3$, $AD = \sqrt { 3 }$. The surface area of sphere $O$ is $13 \pi$. The length of segment $PA$ is \_\_\_\_.

If ( $2,3,5$ ) is one end of a diameter of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 6 x - 12 y - 2 z + 20 = 0$, then the coordinates of the other end of the diameter are
(1) $( 4,9 , - 3 )$
(2) $( 4 , - 3,3 )$
(3) $( 4,3,5 )$
(4) $( 4,3 , - 3 )$

In a spatial coordinate system, there is a globe with center at $O ( 0,0,0 )$ and north pole at $N ( 0,0,2 )$. A point $A$ on the sphere has coordinates $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } , \sqrt { 3 } \right)$. The point on the equator farthest from point $A$ is point $P$. On the great circle passing through points $A$ and $P$, the length of the minor arc between these two points is (blank). (Express as a fraction in lowest terms)

For research laboratories planned to be established on the Luna planet, two completely closed buildings with the same radii and volumes are designed to be placed on the ground as shown in the figure: one in the shape of a half right circular cylinder and the other in the shape of a hemisphere.
Accordingly, what is the ratio of the surface area of the half right circular cylinder building (excluding the ground) to the surface area of the hemisphere building (excluding the ground)?
A) $\frac{1}{2}$ B) $\frac{3}{5}$ C) $1$ D) $\frac{7}{6}$ E) $\frac{4}{3}$