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?
45 m
30 m
103.8 m
94.6 m
Explanation:
Let BD be the lighthouse and A and C be the two points on ground.
Then, BD, the height of the lighthouse = 60 m
$\angle$ BAD = 45° , $\angle$ BCD = 60°
tan45°=$\dfrac{BD}{BA}$
=>1=$\dfrac{60}{BA}$
=>BA=60 m ⋯(1)
tan60°=$\dfrac{BD}{BC}$
=>√3=$\dfrac{60}{BC}$
=>BC=$\dfrac{60}{\sqrt{3}}=\dfrac{60 \times \sqrt{3} }{\sqrt{3} \times \sqrt{3}}=\dfrac{60\sqrt{3} }{3}$
=20√3=20×1.73=34.6 m ⋯(2)
Distance between the two points A and C
= AC = BA + BC
= 60 + 34.6 [∵ Substituted value of BA and BC from (1) and (2)]
= 94.6 m
Let BD be the lighthouse and A and C be the two points on ground.
Then, BD, the height of the lighthouse = 60 m
$\angle$ BAD = 45° , $\angle$ BCD = 60°
tan45°=$\dfrac{BD}{BA}$
=>1=$\dfrac{60}{BA}$
=>BA=60 m ⋯(1)
tan60°=$\dfrac{BD}{BC}$
=>√3=$\dfrac{60}{BC}$
=>BC=$\dfrac{60}{\sqrt{3}}=\dfrac{60 \times \sqrt{3} }{\sqrt{3} \times \sqrt{3}}=\dfrac{60\sqrt{3} }{3}$
=20√3=20×1.73=34.6 m ⋯(2)
Distance between the two points A and C
= AC = BA + BC
= 60 + 34.6 [∵ Substituted value of BA and BC from (1) and (2)]
= 94.6 m