Question

if two towers of height h1 and h2 subtend angle 60dgree and 30dgree at the midpoint of line joining their bases of towers then h1 bata h2 is

Answer 1

Solution -
Hi,
Draw the rough diagram with the data
given 
AI ||| h1 lDI ll lh2B________P_______Cx xJoin A to P and D to C
Height of the one tower = AB = h1
Height of the second tower = CD = h2
Two towers subtend angles of 60 and 30
degrees at P , and P is the mid point of BC
Let BP = PC = x 
i ) In triangle APB , angle B = 90
tan < APB = AB / BP
tan 60 = h1 / x 
( sqrt 3 ) = h1 / x ------( 1 )
ii ) In triangle DCP ,
Angle C = 90,
tan tan 30 = h2 / x
1 / ( sqrt 3 ) = h2 / x -------( 2 )
Find the ratio of ( 1 ) and ( 2 )
Therefore,
(Sqrt 3 ) / [ 1/ sqrt 3 ] = (h1 /x ) / ( h2 / x )
(Sqrt 3 )(sqrt 3 ) / 1 = h1 / h2 
3 /1 = h1 / h2 
Therefore,
h1 : h2 = 3:1I hope this helps you.

:)

Answer 2

$\tan(60) = \frac{h1}{x} \\ \sqrt{3} x = h1 \\ \tan(30) = \frac{h2}{x} \\ \frac{1}{ \sqrt{3} } = \frac{h2}{x} \\ \frac{x}{ \sqrt{3} } = h2 \\ \frac{h1}{h2} = \frac{ \sqrt{3} x}{ \frac{x}{ \sqrt{3} } } = \frac{3}{1}$