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dybincka [34]
3 years ago
9

Dustin is standing at the edge of a vertical cliff, 40 meters high, which overlooks a clear lake. He spots a fluffy white cloud

above the lake, which from his point of view has an angle of elevation of $30^\circ.$ He also sees the reflection of the cloud in the lake, which has an angle of depression of $60^\circ.$ Find the height of the cloud above the lake, in meters.

Mathematics
1 answer:
Licemer1 [7]3 years ago
3 0

Answer:

53.33 meters

Step-by-step explanation:

Let AB represents the height of the cliff,

( where, A is top and B is bottom ),

Also, C and D represents the shadow of the cloud and cloud in the sky respectively,

Suppose E is a point in the segment CD,

Such that,

AB = DE = 40 meters,

According to the question,

m\angle CAE = 30^{\circ}

m\angle EAD = 60^{\circ}

Since,

\tan =\frac{\text{Perpendicular}}{\text{Base}}

\implies \tan 60^{\circ}=\frac{DE}{AE}

\sqrt{3}=\frac{40}{AE}

\implies AE = \frac{40}{\sqrt{3}}

Now,

\tan 30^{\circ}=\frac{CE}{AE}

\frac{1}{\sqrt{3}}=\frac{\sqrt{3}CE}{40}

\implies CE = \frac{40}{3}

Hence,

The height of the cloud above the lake = CE + ED

=\frac{40}{3}+40=13.33+40 = 53.33\text{ meters}

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