(a) 2000 /\(\sqrt3\) m.
Let AB be the diameter of the circular base of the cliff.
Let C be the top of the cliff.
Given,
CE = 500 m , ∠CAE = 60°, ∠CBE = 30°,
AE = d1, BE = d2
In Δ AEC,
tan 60° = \(\frac{500}{d_1}\)
⇒ d1 = \(\frac{500}{\sqrt3}\) m
In Δ BEC,
tan 30° = \(\frac{500}{d_2}\)
⇒ d2 = 500\(\sqrt3\) m
∴ Required diameter = d1 + d2
= \(\frac{500}{\sqrt3}\) + 500\(\sqrt3\) = \(\frac{2000}{\sqrt3}\) m