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November 3rd, 2016, 07:33 AM  #1 
Newbie Joined: Nov 2016 From: Gujarat Posts: 3 Thanks: 1  Height and distance
From top of a tower the angle of depression of the top and bottom of a multistoreyed building are 30° and 60° respectively. If the height of the building is 100m, find the height of the tower.
Last edited by skipjack; November 3rd, 2016 at 07:42 AM. 
November 3rd, 2016, 08:13 AM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 3,005 Thanks: 1588 
see attached sketch h = tower height x = horizontal distance between tower and building solve the system of equations for h ... 
November 3rd, 2016, 08:30 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,942 Thanks: 2210 
By the angle bisector theorem, (h  100m)/100m = x/h = cos(60°) = 1/2, where x and h are as in skeeter's diagram, so h = 150m.

November 3rd, 2016, 09:44 AM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond 
A triangle is 306090, $(2x)^2=x^2+h^2\implies3x^2=h^2\implies x=\frac{h}{\sqrt3}$. Substitute for $x$ in the equations to the right of the diagram and subtract: $$100=\frac{h}{\sqrt3}(\tan60\tan30)=h\left(1\frac13\right)\implies h=150\text{ m}$$ 
November 3rd, 2016, 12:26 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,942 Thanks: 2210 
The 30°30°120° triangle has shorter sides both of length 100m. As the small 30°60°90° triangle above it is half of an equilateral triangle, h  100m = (100m)/2, and so h = 150m. 
November 4th, 2016, 05:00 PM  #6 
Newbie Joined: Nov 2016 From: MI Posts: 4 Thanks: 2 Math Focus: Trigonometry and Precalculus. Some occasional calc 1/2 fun.  
November 4th, 2016, 10:50 PM  #7 
Newbie Joined: Nov 2016 From: Gujarat Posts: 3 Thanks: 1 
Thanks

November 5th, 2016, 03:39 AM  #8 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond 
Using $3x^2=h^2$ and similar triangles (from skeeter's diagram), $$\frac{h100}{x}=\frac xh$$ $$h^2100h=x^2$$ $$3h^2300h=3x^2=h^2\implies h=150$$ 

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