Question

As observed from the top of a light house, 100m above sea level, the angle of depression of a ship, sailing directly towards it, changes from 30 degree to 45 degree. Determine the distance traveled by the ship during the period of observation.

Answer 1

30 + 45 =75
100 - 75 =25

Answer 2

$\huge\underline\mathfrak{Diagram:-}$

$\underline\mathfrak{Given:-}$

• $\: \: \: \: \: \: \: Height \: \: of \: \: of \: \: light \: \: house \: \: = \: \: {100m}$

• $\: \: \: \: \: \: \: Angle \: \: of \: \: depression \: \: of \: \: {1st} \: \: ship \: \: = \: \: {45} \degree$

• $\: \: \: \: \: \: \: Angle \: \: of \: \: depression \: \: of \: \: {2nd} \: \: ship \: \: = \: \: {30} \degree$

$\underline\mathfrak{To \: \: Find:-}$

• $\: \: \: \: \: Distance \: \: between \: \: two \: \: ship?$

$\underline\mathfrak{Solutions:-}$

• $\: \: \: \: \: \: \: AB \: \: = \: \: {100m}$

• $\: \: \: \: \: \angle ACB \: \: = \: \: {45} \degree$

• $\: \: \: \: \: \angle ADB \: \: = \: \: {30} \degree$

$\: \: \: \: \: \therefore \: In \: \: \triangle \: ABC.$

$\: \: \: \: \: \leadsto \frac{100}{BC} \: \: = \: \: tan {45} \degree$

$\: \: \: \: \: \leadsto {BC} \: \: = \: \: {100m}$

$\: \: \: \: \: \therefore \: In \: \: \triangle \: ABD.$

$\: \: \: \: \: \leadsto \frac{100}{BC \: + \: x} \: \: = \: \: tan {30} \degree$

$\: \: \: \: \: \leadsto {100} \: \: = \: \: \frac{1}{\sqrt{3}} \: \times \: {({100} \: + \: x)}$

$\: \: \: \: \: \leadsto {{100}\sqrt{3}} \: \: = \: \: {100} \: + \: x$

$\: \: \: \: \: \leadsto {{100}\sqrt{3}} \: - \: {100} \: \: = \: \: x$

$\: \: \: \: \: \leadsto {100} \: {({\sqrt{3}} \: - \: {1})} \: \: = \: \: x$

$\: \: \: \: \: \leadsto {100} \: {({1.732} \: - \: {1})} \: \: = \: \: x$

$\: \: \: \: \: \leadsto {100} \: \times \: {0.732} \: \: = \: \: x$

$\: \: \: \: \: \leadsto {100} \: \times \: \frac{732}{1000} \: \: = \: \: x$

$\: \: \: \: \: \leadsto \frac{732}{10} \: \: = \: \: x$

$\: \: \: \: \: \leadsto {x} \: \: = \: \: {73.2m}$

• $\: \: \: \: \: Hence, \: \: distance \: \: between \: \: two \: \: {73.2m}$

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