Question

The tops of two towers of height x and y, standing on level ground, subtend angles of 30° and 60° respectively at the centre of the line joining their feet, then find x: y.

Answer 1

Answer:

The ratio of towers height x : y is 1 : 3  

Step-by-step explanation:

Given as :

The height of tower one = x meters

The height of tower two = y meters

The angle of depression of x m high tower = 60°

The angle of depression of y m high tower = 30°

The tower are standing on level ground

The distance of base of both tower to their meet point o = a meters

According to question

From figure

In Δ COD

Tan angle = $\dfrac{perpendicular}{base}$

Or, Tan 30° = $\dfrac{CD}{OC}$

Or, $\dfrac{1}{\sqrt{3} }$ = $\dfrac{x}{a}$

∴   a = √3 x                ...........1

 So, The distance of point O from tower base = OC = a = √3 y meters    

Again

In Δ AOB

Tan angle = $\dfrac{perpendicular}{base}$

Or, Tan 60° = $\dfrac{AB}{OA}$

Or, √3 = $\dfrac{y}{a}$

∴   a =  $\dfrac{y}{\sqrt{3} }$         .........2   

So, The distance of point O from tower base = AO = a = $\dfrac{a}{\sqrt{3} }$ meters  

Now, From eq 1 and eq 2

√3 x   =   $\dfrac{y}{\sqrt{3} }$

Or,  y = 3 x

∴  $\dfrac{x}{y}$ = $\dfrac{1}{3}$

So, The ratio of their height =   $\dfrac{x}{y}$ = $\dfrac{1}{3}$

Hence, The ratio of towers height x : y is 1 : 3  Answer