1.26: Figure 3 shows a right pyramid with a horizontal square base.

4PM1/1/November/2023 — Question 6 · 11 marks

1.26 diagram 1
Figure 3 shows a right pyramid with a horizontal square base.
AB=BC=CD=DA=xcmAB = BC = CD = DA = x \mathrm{cm}
AV=BV=CV=DV=xcmAV = BV = CV = DV = x \mathrm{cm}
O\displaystyle O is the point of intersection of the diagonals of the base.
The vertex V\displaystyle V of the pyramid is vertically above O\displaystyle O
(a) Show that VO=22xcm\displaystyle VO = \frac{\sqrt{2}}{2}x \mathrm{cm}
(3)
(b) Find, in degrees, the size of the angle AVC\displaystyle AVC
(2)
(c) Find, in degrees to one decimal place, the size of the angle between the plane VAB\displaystyle VAB and the plane VDC\displaystyle VDC
(3)
The volume of the pyramid is 200cm3\displaystyle 200 \mathrm{cm}^3
Given that the volume of a pyramid =13×base area×height\displaystyle = \frac{1}{3} \times \text{base area} \times \mathrm{height}
(d) Find to 3 significant figures, the value of x\displaystyle x
(3)