Name: Introduction to Heights and Distances
Uploaded: 2019-07-19
Duration: 4 min 50 s
Description: Let's take a look at the basics of trigonometry through this introduction to heights and distances. Learn about the angle of elevation and the angle of depre...

Question

From the top of a cliff, the angle of depression of the top and bottom of a tower are observed to be

45 °

and

60 °

respectively. If the height of the tower is

20 m

, find the height of the cliff.

Accepted Answer

Let, $A B$ is the height of tower $= 20 m$ and $C E$ is the height of the cliff.

Now, $C E = C D + D E$ and $C D = 20 m$

Let $E D = x m$ and the distance between the cliff and the tower $=$ $B C = y m$

In $∆ A E D$ ,

$\tan 45 ° = \frac{E D}{A D} \Rightarrow 1 = \frac{x}{y} \Rightarrow x = y$

Similarly, in $∆ B C E,$

$\frac{B C}{C E} = \tan 60 ° \Rightarrow \frac{20 + x}{y} = \sqrt{3} \Rightarrow \frac{20 + x}{x} = \sqrt{3} \Rightarrow 20 + x = \sqrt{3} x \Rightarrow x (\sqrt{3} - 1) = 20 \Rightarrow x = \frac{20}{(\sqrt{3} - 1)} \Rightarrow x = \frac{20 (\sqrt{3} + 1)}{(\sqrt{3} - 1) (\sqrt{3} + 1)} \Rightarrow x = \frac{20 (\sqrt{3} + 1)}{3 - 1} \Rightarrow x = \frac{20 (\sqrt{3} + 1)}{2} \Rightarrow x = 10 (\sqrt{3} + 1)$

$\Rightarrow x = 10 (1.73 + 1) \Rightarrow x = 10 \times 2.73 \Rightarrow x = 27.3$

Now, we have

$x = y \Rightarrow y = 27.3 m (approx)$

Hence, the height of the cliff $= 20 + x = 47.3 m$ (approx).

From the top of a cliff, the angle of depression of the top and bottom of a tower are observed to be $45 °$ and $60 °$ respectively. If the height of the tower is $20 m$ , find the height of the cliff.

Important Questions on Heights and Distances

Learn and Prepare for any exam you want !

From the top of a cliff, the angle of depression of the top and bottom of a tower are observed to be 45° and 60° respectively. If the height of the tower is 20 m, find the height of the cliff.

Important Questions on Heights and Distances

Learn and Prepare for any exam you want !

From the top of a cliff, the angle of depression of the top and bottom of a tower are observed to be $45 °$ and $60 °$ respectively. If the height of the tower is $20 m$ , find the height of the cliff.