maths mock 問題1條

b)(i)
Hint:
Volume of tetrahedron = (1/3)(base area)(height)
since the base area (area of ΔBCD) is fixed,
the height has to be maximum in order to achieve the maximum volume
ie, plane BCD and plane ABD must be perp. to each other.
in other words, if O is a point on BD such that AO perp. to BD
then AO will be the height of the tetrahedron

BD = 10 (pyth. thm) (part a應該計左BD長度，略解)
sin∠DBA = AD/DB = 8/10 = 4/5
sin∠DBA = AO/AB
4/5 = AO/6
AO = 24/5 = 4.8

Area of ΔBCD = 16.583 cm² (part a應該計左, 略解)
Volume of tetrahedral
= (1/3)(base area)(height)
= (1/3)(16.583)(4.8)
= 26.5cm³ --##

(ii)
Volume of tetrahedron = (1/3)(base area)(height)
in order to achieve half of the max. vol., height should be half of AO.
let h be the corresponding height,
θ be the angle between planes ABD and CBD
h = AO cosθ
for 0° < θ < 90°, cosθ = 1/2 when θ=30°
so such volume will be attained when θ=30°,
ie, Alan's claim is correct.