A hall has a 10 m10 m square floor and vertical walls.

Question

A hall has a $10	ext m	imes10	ext m$ square floor and vertical walls. The angle $\angle GPH$ between diagonals $AG$ and $BH$ is $\cos^{-1}\dfrac15$. The height of the hall (in m) is:

Accepted Answer

$A=(0,0,0)$, $B=(10,0,0)$, $G=(10,10,h)$, $H=(0,10,h)$: $\vec{AG}=(10,10,h)$, $\vec{BH}=(-10,10,h)$.

$$\cos\theta = \frac{\vec{AG}\cdot\vec{BH}}{|\vec{AG}||\vec{BH}|} = \frac{-100+100+h^2}{h^2+200} = \frac{h^2}{h^2+200} = \frac15$$
$$5h^2 = h^2+200 \implies h^2=50 \implies h=5\sqrt2$$

Solutions · 1