The diagram shows the structure of a building with a horizontal rectangular base $ABCD$, whereby \

Question

The diagram shows the structure of a building with a horizontal rectangular base $ABCD$, whereby $BC$ is $m$ units.

The structure consists of four vertical columns, $AE$, $DH$, $CG$, and $BF$ which are parallel to the $z$- axis.

The roof $EFGH$ has equation $x-y+\sqrt{2}z=2+4\sqrt{2}$ and the plane $ABFE$ has equation $x+y=2$. The origin $(0,0,0)$ is a point within the rectangular base $ABCD$.

Find the cartesian equation of the plane $DCGH$ in terms of $m$.[2]
Find the acute angle between the roof $EFGH$ and the base $ABCD$.[2]

To ensure that the roof is sturdier, a beam needs to be added to connect point $W(0,2,4)$, which lies on column $AE$, to the nearest point on the roof.

Show that this nearest point of the roof is $\left( 1,1,4+\sqrt{2} \right)$.[3]
Hence find the exact cartesian equation of the beam $EF$.[3]

Accepted Answer

(i) ${x + y = 2 - m\sqrt{2}	ext{ , where } m > \sqrt{2}}$ (ii) ${45^\circ}$ (iii) ${N(1,1,4+\sqrt{2})}$ (iv) ${x - 1 = 1 - y = \frac{4 + \sqrt{2} - z}{\sqrt{2}}}$

2018 ACJC MYE Q7

2018 ACJC MYE Q7

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