1.36: Past-paper question 10

4PM1/2R/June/2024 — Question 10 · 13 marks

1.36 diagram 1
Figure 4 shows a right pyramid ABCDV\displaystyle ABCDV
The base of the pyramid is a rectangle where,
AB=DC=2xcmAD=BC=xcmAB = DC = 2x \mathrm{cm} \quad AD = BC = x \mathrm{cm}
The edges VA\displaystyle VA, VB\displaystyle VB, VC\displaystyle VC and VD\displaystyle VD are all of equal length.
The angle between VA\displaystyle VA and ABCD\displaystyle ABCD is 45∘\displaystyle 45^\circ
(a) Show that VA=102x\displaystyle VA = \frac{\sqrt{10}}{2}x cm
(3)
(b) Find in cm, the exact height of the pyramid in terms of x\displaystyle x
(2)
Find, in degrees to one decimal place,
(c) the size of angle VBA\displaystyle VBA
(2)
(d) the size of the obtuse angle between the plane AVC\displaystyle AVC and the plane BVD\displaystyle BVD
(4)
Given that the volume of the pyramid is 95\displaystyle 9\sqrt{5}cm3\displaystyle \mathrm{cm}^{3}
(e) find the value of x\displaystyle x
(2)