Given is the sphere with center $M ( 1 | 4 | 0 )$ and radius 6.
(1a) [3 marks] Determine all values $p \in \mathbb { R }$ for which the point $P ( 5 | 1 | p )$ lies on the sphere.
(1b) [2 marks] The line $g$ is tangent to the sphere at the point $B ( - 3 | 8 | 2 )$. Find a possible equation of $g$.
For each value of $a$ with $a \in \mathbb { R }$, a line $g _ { a }$ is given by $g _ { a } : \vec { X } = \left( \begin{array} { c } 2 \\ a - 4 \\ 4 \end{array} \right) + \lambda \cdot \left( \begin{array} { c } 2 \\ - 2 \\ 1 \end{array} \right) , \lambda \in \mathbb { R }$
(2a) [2 marks] Determine, depending on $a$, the coordinates of the point where $g _ { a }$ intersects the $x _ { 1 } x _ { 2 }$ plane.
(2b) [3 marks] For exactly one value of $a$, the line $g _ { a }$ has an intersection point with the $x _ { 3 }$-axis. Find the coordinates of this intersection point.
On a playground, a triangular sun sail is erected to shade a sandbox. For this purpose, metal poles are fixed in the ground at three corners of the sandbox, at whose ends the sun sail is fastened. In a Cartesian coordinate system, the $x _ { 1 } x _ { 2 }$-plane represents the horizontal ground. The sandbox is described by the rectangle with corner points $K _ { 1 } ( 0 | 4 | 0 ) , K _ { 2 } ( 0 | 0 | 0 ) , K _ { 3 } ( 3 | 0 | 0 )$ and $K _ { 4 } ( 3 | 4 | 0 )$. The sun sail is represented by the planar triangle with corner points $S _ { 1 } ( 0 | 6 | 2,5 ) , S _ { 2 } ( 0 | 0 | 3 )$ and $S _ { 3 } ( 6 | 0 | 2,5 )$ (see Figure 1). One unit of length in the coordinate system corresponds to one meter in reality.
[Figure]The three points $S _ { 1 } , S _ { 2 }$ and $S _ { 3 }$ determine the plane $E$. Sub-task Part B a (4 marks) Find an equation of the plane $E$ in normal form. (for verification: $E : x _ { 1 } + x _ { 2 } + 12 x _ { 3 } - 36 = 0$ ) Sub-task Part B b (3 marks) The manufacturer of the sun sail recommends stabilizing the metal poles used with additional safety cables if the sun sail area is more than $20 \mathrm {~m} ^ { 2 }$. Assess whether such stabilization is necessary in the present situation based on this recommendation.
Sunrays fall on the sun sail, which in the model and in Figure 1 can be represented by parallel lines with direction vector $\overrightarrow { S _ { 1 } K _ { 1 } }$. The sun sail casts a triangular shadow on the ground. The shadows of the corners of the sun sail designated by $S _ { 2 }$ and $S _ { 3 }$ are designated by $S _ { 2 } ^ { \prime }$ and $S _ { 3 } ^ { \prime }$ respectively.
Sub-task Part B c (2 marks) Justify without further calculation that $S _ { 2 } ^ { \prime }$ lies on the $x _ { 2 }$-axis. Sub-task Part B d (3 marks) $S _ { 3 } ^ { \prime }$ has the coordinates $( 6 | - 2 | 0 )$. Draw the triangle representing the shadow of the sun sail in Figure 1. Decide from the drawing whether more than half of the sandbox is shaded.
Sub-task Part B e (3 marks) To ensure the drainage of rainwater, the sun sail must have an inclination angle of at least $8 ^ { \circ }$ with respect to the horizontal ground. Justify that the drainage of rainwater is not ensured in the present case.
Sub-task Part B f (5 marks) In heavy rain, the sun sail deforms and sags. A so-called water pocket forms from rainwater that cannot drain away. The top surface of the water pocket is horizontal and is approximately circular with a diameter of 50 cm. At its deepest point, the water pocket is 5 cm deep. For simplicity, the water pocket is considered as a spherical segment (see Figure 2).
[Figure]Fig. 2The volume $V$ of a spherical segment can be calculated using the formula $V = \frac { 1 } { 3 } \pi h ^ { 2 } \cdot ( 3 r - h )$, where $r$ denotes the radius of the sphere and $h$ denotes the height of the spherical segment. Determine how many liters of water are in the water pocket.