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August 26th, 2015, 12:17 AM  #1 
Newbie Joined: Oct 2012 Posts: 29 Thanks: 0  Finding the minimum length of l using trig functions
The Problem given was that there's a house of unimportant height and wall 10ft tall 8ft from the house. Now, a ladder is placed such that one end of the ladder rests on a point on the wall of the house and the base rests outside the wall. A figure is given (see attachment). The left line represents the house. If there are points that aren't clear, feel free to ask. Any explanation is greatly appreciated. Last edited by skipjack; August 26th, 2015 at 01:14 PM. 
August 26th, 2015, 02:34 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,805 Thanks: 2150 
What are you asked to do or to explain?

August 26th, 2015, 06:41 AM  #3 
Newbie Joined: Oct 2012 Posts: 29 Thanks: 0 
It's in the title. We have to find the minimum length of the ladder using trigonometric functions. 
August 26th, 2015, 08:54 AM  #4  
Senior Member Joined: Feb 2010 Posts: 706 Thanks: 141  Quote:
 
August 26th, 2015, 10:20 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,805 Thanks: 2150 
Are you sure?

August 26th, 2015, 10:53 AM  #6 
Senior Member Joined: Feb 2010 Posts: 706 Thanks: 141  
August 26th, 2015, 12:15 PM  #7 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,949 Thanks: 1141 Math Focus: Elementary mathematics and beyond 
Using similar triangles and a graphing calculator, I get $l$ $\approx$ 25.4033 feet. Volle, did you have a particular approach in mind? 
August 26th, 2015, 01:06 PM  #8 
Global Moderator Joined: Dec 2006 Posts: 20,805 Thanks: 2150 
Let θ be the acute angle of elevation of the ladder, then its length l = (8sec θ + 10csc θ) ft. To locate the minimum, we solve dl/dθ = 0. We need 8sec θ tan θ  10cot θ csc θ = 0, so tan³ θ = 5/4. Plugging the appropriate value of θ into the equation for l gives l = 25.4 ft after rounding to 3 significant figures. An exact expression can be found for this, but is inelegant. 
August 26th, 2015, 03:03 PM  #9  
Newbie Joined: Oct 2012 Posts: 29 Thanks: 0  Quote:
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Tags 
finding, functions, length, minimum, trig 
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