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Geometry of using a ladder?

Posted on 2012-12-20
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Last Modified: 2012-12-26
It's been a long time since I used geometry.

Extension ladder:
1. It is imperative that you understand that an extension ladder must extend 3 feet above the top landing.
3. The correct angle for an extension ladder should be ¼, meaning that for every 3-4 feet up, the ladder should be 1 foot out away from the base.
6. The sections of the extension ladder must overlap by at least 3 feet.
http://nationalsafety.wordpress.com/2010/04/20/the-basics-of-extension-ladder-safety/

So, a 28 foot two piece extension ladder, would be about 7 feet from the base.
28 / 4 = 7.

A. What is the angle of the ladder in relation to perpendicular (straight up which would be 90 degrees)
B. If the two sections need to overlap 3 feet, and the top needs to be 3 feet above the landing, and the angle needs to be "x degrees from perpendicular," how high can this 28 foot ladder reach?

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I tried to draw a diagram using Paint but failed.

So, 7 feet from bottom of ladder to the building.
Top of ladder 3 feet above the top edge of building.
What is the angle and what is the height able to be reached?
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Question by:nickg5
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d-glitch earned 680 total points
ID: 38711448
The angle of the ladder is the arctangent of 4/1 = 75.9 degrees.

>> an extension ladder must extend 3 feet above the top landing.
This is if you have to step off the top of the ladder on to a roof or landing.

The top of any ladder at 75.9 degrees is  
     length x sin( 75.9)      or     length * 0.97
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by:d-glitch
ID: 38711455
If both sections are 14 ft each, you really only have 25 ft ladder.
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by:nickg5
ID: 38711457
28 x .97 = 27.16 feet.

That does not allow for the two sections to overlap 3 feet.
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by:nickg5
ID: 38711461
If 3 feet over the top and 3 feet overlap of the two sections, we're down to 22 feet before the angle.
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by:d-glitch
ID: 38711486
Yes, and the angle costs you 3%  or 7.9 inches.
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Assisted Solution

by:dhsindy Sparrow
dhsindy Sparrow earned 660 total points
ID: 38711493
The angle at the building is 14 degrees at the ground 76 degrees.

Arctangent (1/4) = 14 degrees (top angle).  The ground angle is (90-14) degrees = 76 degrees.  Accurate to within about 0.04 degrees plus or minus.

|\
|  \                         the hypotnuse is sqrt of 17
|    \     4 vertical units
|____\  ground angle

     1 horizontal unit
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by:dhsindy Sparrow
ID: 38711500
The angles remain the same no matter the size of the triangle.  Maybe these are called congruent trangles I forget.  Use trig, easier for angles.
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by:dhsindy Sparrow
ID: 38711510
Note:  Oops, I didn't refresh before submitting.  My apologies to d-glitch.
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by:nickg5
ID: 38711519
If each unit is 1 foot, then your example is a 4 foot ladder (but using 16 for easy math = square root of 16 = 4)
The hypotenuse at 76 degrees would be 4 feet.
Since each unit is 1 foot then it's a 4 foot ladder, but it should lose 3% due to the angle.

Not sure if a 28 foot ladder would be 28 units. Yes, I think.
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by:dhsindy Sparrow
ID: 38711532
Being as I have worked as a safety engineer before, I must warn you to check the capacity tag on the side of any ladder you use.  Most are rated for 250 lbs which is not enough for me.  I have to buy construction rated ladders for big people carrying materials.
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Assisted Solution

by:awking00
awking00 earned 660 total points
ID: 38714042
You don't really need to know the angle to determine the maximum height a 28 ft. ladder can reach. Knowing you need a 3 ft. extension means the hypoteneuse will be 25 ft. so
25^2 = x^2 + (x/4)^2 = x^2 + x^2/16 ==> multiplying both sides by 16
16*25^2 = 17*x^2
16*625 = 17*x^2
10000 = 17*x^2 ==> dividing both sides by 17
10000/17 = x^2
588.24 = x^2 ==> taking the square root of both sides
24.25 = x  ==> the maximum height the ladder will reach
6.125 ==> is the distance the base of the ladder will be from the wall
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by:nickg5
ID: 38721473
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