A drone in a warehouse travels in a straight line through points $C(-1,3,6)$ and $D(1,2,8)$, measured from the corner $O$ of the building. The plane $y=0$ represents a side wall and the $z$-axis points vertically upwards. Distances are given in metres. The drone continues along the same path until it meets the wall at point $Q$.

A laser beam in a gallery travels in a straight line through points $A(4,-2,5)$ and $B(1,4,2)$, measured from the corner $O$ of the room. The $xy$-plane represents the floor and the $z$-axis points vertically upwards. Distances are given in metres. The beam continues until it meets the floor at point $P$.

A straight line, $d_1$, is given by the equation $r=\left(\begin{array}{c}6\\ 1\\ -4\end{array}\right)+\mu \left(\begin{array}{c}-3\\ 2\\ 5\end{array}\right)$. Point $A(p,9,16)$ lies on $d_1$.

The points $A(4,-1,6)$ and $B(7,3,0)$ lie on line $d_1$. A second line $d_2$ is given by $\mathbf{r}=\left(\begin{array}{c}16\\ 4\\ q\end{array}\right)+\mu \left(\begin{array}{c}-2\\ 1\\ 5\end{array}\right)$, where $q$ is a constant. It is known that $d_1$ and $d_2$ intersect at a point $T$.

An inspection drone in a warehouse has position coordinates $(x,y,z)$, where $x$ and $y$ are the horizontal displacements from the centre of a charging pad on the warehouse floor and $z$ is the height above the floor. All distances are in metres.

At 14:30, the drone is at $(15,-20,36)$. At 14:32, it is at $(9,-12,28)$. Let $t$ be the time in minutes after 14:30.

Assume the drone continues to move in a straight line with constant velocity.

When it reaches a height of $24$ m, it keeps the same horizontal motion but changes its rate of descent so that it arrives exactly at $(0,0,0)$.