The angle of elevation of the top of a lighthouse 60 m high, from two points on the ground on its opposite sides are 45° and 60°. What is the distance between these two points?

A. 45 m B. 30 m C. 103.8 m D. 94.6 m Answer: Option D
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Solution(By Apex Team)

Let BD be the lighthouse and A and C be the two points on ground. Then, BD, the height of the lighthouse = 60 m angle of elevation height and distance. image $\begin{aligned}\angle BAD&=45^{\circ},\angle BCD=60^{\circ}\\ \tan45^{\circ}&=\frac{BD}{BA}\\ \Rightarrow1=&\frac{60}{BA}\\ \Rightarrow BA&=60\mathrm{~m}\ldots\ldots\ldots(\mathrm{i})\\ \tan60^{\circ}&=\frac{BD}{BC}\\ \Rightarrow\sqrt{3}&=\frac{60}{BC}\\ \Rightarrow BC&=\frac{60}{\sqrt{3}}\\ &=\frac{60\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}\\ &=\frac{60\sqrt{3}}{3}\\ &=20\sqrt{3}\\ &=20\times1.73\\ &=34.6\mathrm{~m}\ldots\ldots\ldots\text{ (ii) }\end{aligned}$ Distance between the two points A and C = AC = BA + BC = 60 + 34.6 [∵ Substituted value of BA and BC from (i) and (ii)] = 94.6 m