A gas company has plans to install a pipeline from a gas field to a storage facility. One part of th

Question

A gas company has plans to install a pipeline from a gas field to a storage facility. One part of the route for the pipeline has to pass under a river. This part of the pipeline is in a straight line between two points, $P$ and $Q$. Points are defined relative to an origin $\left( 0,0,0 \right)$ at the gas field. The $x$-, $y$- and $z$- axes are in the directions east, north and vertically upwards respectively, with units in metres. $P$ has coordinates $\left( 1136,\,\,92,\,\,p \right)$ and $Q$ has coordinates $\left( 200,\,\,20,\,\,-15 \right)$. The length of the pipeline $PQ$ is $939$ m. Given that the level of $P$ is below that of $Q$, find the value of $p$. [3] A thin layer of rock lies below the ground. This layer is modelled as a plane. Three points in this plane are $\left( 400,\,600,\,-20 \right)$, $\left( 500,\,200,\,-70 \right)$ and $\left( 600,\,-340,\,-50 \right)$. Find the cartesian equation of this plane.Find the cartesian equation of this plane. [4] Hence find the coordinates of the point where the pipeline meets the rock. [4] Find the angle that the pipeline between the points $P$ and $Q$ makes with the horizontal. [2]

Accepted Answer

(a) $p=6$ (rejected $\because p<-15$) 	ext{ or } $-36$ (b) $5x+y+2z=2560$ (c) $\left( 512,\,44,\,-22 ight)$ (d) $	heta ={{1.3}^{\circ }}$ ($1$ d.p.)

N2022 P1 Q11

N2022 P1 Q11

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