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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
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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, 08:30 AM #3 Global Moderator   Joined: Dec 2006 Posts: 20,631 Thanks: 2077 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, 09:44 AM #4 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,935 Thanks: 1129 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, 12:26 PM #5 Global Moderator   Joined: Dec 2006 Posts: 20,631 Thanks: 2077 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, 05:00 PM   #6
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Math Focus: Trigonometry and Precalculus. Some occasional calc 1/2 fun.
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, 10:50 PM #7 Newbie   Joined: Nov 2016 From: Gujarat Posts: 3 Thanks: 1 Thanks Thanks from greg1313
 November 5th, 2016, 03:39 AM #8 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,935 Thanks: 1129 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

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