1.43: Past-paper question 8

Figure 3 shows a right pyramid with vertex $\displaystyle V$ and square base, $\displaystyle ABCD$, of side 12 cm

The size of the angle between the plane $\displaystyle VAB$ and the base $\displaystyle ABCD$ is $\displaystyle 30^\circ$

(a) Show that the height of the pyramid is $\displaystyle 2\sqrt{3}$ cm

(b) Find, in cm, the exact length of $\displaystyle VA$

(c) Find, in $\displaystyle \mathrm{cm}^2$, the exact area of triangle $\displaystyle VAD$

(d) Find, in cm, the exact length of the perpendicular from $\displaystyle D$ to $\displaystyle VA$

(e) Find, in degrees to one decimal place, the size of the obtuse angle between the plane $\displaystyle VAB$ and the plane $\displaystyle VAD$