(a) In the diagram, (Delta) ABD is right-angled at B. |AB| = 3 cm, |AD| = 5 cm, (stackrelfrown{ACB}) = 61° and (stackrelfrown{DAC}) = x°. Calculate, correct to one decimal place, the value of x.
(b) In the diagram, OABCD is a pyramid with a square base of side 2cm and a slant height of 4 cm. Calculate, correct to three significant figures : (i) the vertical height of the pyramid ; (ii) the volume of the pyramid.
Explanation
(a) In (Delta ACB, < CAB = 180° – (90° + 61°))
= (180° – 151°)
= (29°)
In (Delta ADB, < DAB = (29 + x)°)
(cos (29 + x) = frac{3}{5} = 0.6)
((29 + x)° = cos^{-1} (0.6))
((29 + x)° = 53.13°)
(x = 53.13° – 29°)
= (24.13°)
(approxeq 24.1°) (1 decimal place).
(b) (i) (DB = 2DN = 2NB)
(OA = OB = OC = OD = 4cm)
(AB = BC = CD = DA = 2cm)
(DB^{2} = BC^{2} + DC^{2})
= (2^{2} + 2^{2})
= (sqrt{8})
= (2sqrt{2} cm)
(2DN = DB implies DN = frac{2sqrt{2}}{2} = sqrt{2} cm)
In (Delta DON),
(OD^{2} = ON^{2} + DN^{2})
(4^{2} = h^{2} + (sqrt{2})^{2} implies h^{2} = 16 – 2)
(h^{2} = 14 implies h = sqrt{14})
(ii) (V = frac{1}{3} times base area times perp distance)
(V = frac{1}{3} times 2^{2} times sqrt{14})
= (frac{4sqrt{14}}{3})
= (4.99 cm^{3})