The diagram above shows an object with $O$ at the centre of its rectangular base $ABCD$ where $AB=8$

Question

The diagram above shows an object with $O$ at the centre of its rectangular base $ABCD$ where $AB=8$cm and $BC=4$cm. The top side of the object, $EFGH$ is a square with side $2$cm long and is parallel to the base. The centre of the top side is vertically above $O$ at a height $h$ cm. Show that the equation of the line $BG$ may be expressed as $r=\left( \begin{matrix}4 \-2 \ 0 \\end{matrix} ight)+t\left( \begin{matrix}-3 \ 1 \ h \\end{matrix} ight)$, where $t$ is a parameter. [1] Find the sine of the angle between the line $BG$ and the rectangular base $ABCD$ in terms of $h$. [2] It is given that $h=6$. Find the cartesian equation of the plane $BCFG$. [3] Find the shortest distance from the point $A$ to the plane $BCFG$. [2] The line $l$, which passes through the point $A$, is parallel to the normal of plane $BCFG$. Given that, the line $l$ intersects the plane $BCFG$ at a point $M$, use your answer in part (iv) to find the shortest distance from point $M$ to the rectangular base $ABCD$. [2]

Accepted Answer

sinβ = h/√(10+h²); Π_BCFG: r·(2,0,1) = 8; 2x+z=8; 16/√5 units; height = 16/5

2019 ASRJC P1 Q9

2019 ASRJC P1 Q9

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