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 November 3rd, 2016, 08:33 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 08:42 AM. November 3rd, 2016, 09:13 AM   #2
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see attached sketch

h = tower height

x = horizontal distance between tower and building

solve the system of equations for h ...
Attached Images tower height.jpg (32.3 KB, 12 views) November 3rd, 2016, 09:30 AM #3 Global Moderator   Joined: Dec 2006 Posts: 21,119 Thanks: 2331 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. Thanks from Neel November 3rd, 2016, 10:44 AM #4 Global Moderator   Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,979 Thanks: 1161 Math Focus: Elementary mathematics and beyond A triangle is 30-60-90, $(2x)^2=x^2+h^2\implies3x^2=h^2\implies x=\frac{h}{\sqrt3}$. Substitute for $x$ in the equations to the right of the diagram and subtract: $$100=\frac{h}{\sqrt3}(\tan60-\tan30)=h\left(1-\frac13\right)\implies h=150\text{ m}$$ Thanks from Neel November 3rd, 2016, 01:26 PM #5 Global Moderator   Joined: Dec 2006 Posts: 21,119 Thanks: 2331 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. Thanks from Neel November 4th, 2016, 06:00 PM   #6
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Quote:
 Originally Posted by skipjack 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.
totally forgot about the angle bisector theorem. November 4th, 2016, 11:50 PM #7 Newbie   Joined: Nov 2016 From: Gujarat Posts: 3 Thanks: 1 Thanks Thanks from greg1313 November 5th, 2016, 04:39 AM #8 Global Moderator   Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,979 Thanks: 1161 Math Focus: Elementary mathematics and beyond Using $3x^2=h^2$ and similar triangles (from skeeter's diagram), $$\frac{h-100}{x}=\frac xh$$ $$h^2-100h=x^2$$ $$3h^2-300h=3x^2=h^2\implies h=150$$ Thanks from Neel Tags distance, height Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Monox D. I-Fly Geometry 10 April 7th, 2016 12:34 AM rakmo Algebra 3 March 28th, 2013 06:20 AM Luke4bs Algebra 2 August 14th, 2012 09:57 AM Joker757 Algebra 1 February 5th, 2012 04:02 PM symmetry Algebra 2 September 11th, 2007 08:20 PM

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