A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?
2006-10-31 11:05:31 · 1 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
That would be 12 feet times the square root of 2, (or1.414...) assuming the ladder is straight and rigid. If it's a rope ladder then the answer is 12 feet. Actually, I have had some more time to think about this and I now believe my first answer was incorrect. The ladder length equals 8sec(theta)+4sec(ninety-theta) ........where theta is the angle of the ladder against the wall of the building. I believe the correct answer is for an angle theta of 30 degrees. To find the correct answer mathematically, you would need to take the first derivative of this equation and find the angle theta where the first derivative is zero, since this will be the point in the curve where the ladder is the smalIest or at a minimum. I hope this helps.
2006-10-31 11:18:12 · answer #1 · answered by Sciencenut 7 · 0⤊ 6⤋
Draw a diagram. You have 2 similar triangles. Let x = distance from fence to bottom of ladder h = height at which ladder hits the building L = length of ladder From similar triangles: h/(x+4) = 8/x L = sq. rt.[ (x+4)^2 + h^2] From the first equation, solve for x in terms of h Substitute in the L equation, and solve for a minimum, by differentiating and setting the result = 0 (I assume this is a calculus problem involving max and min)
2016-03-19 02:17:51 · answer #2 · answered by Anonymous · 0⤊ 0⤋
The shortest distance ladder will always be formed by a 45-degree angle to the ground.
The 8' tall fence requires the ladder to touch the ground 8' from the fence. Since the building is 4' further back, the total ground length is 12'.
Therefore the ladder is 12*SQRT(2) or a little under 17' long.
2006-10-31 11:23:55 · answer #3 · answered by Steve A 7 · 4⤊ 6⤋
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2015-01-24 09:00:16 · answer #4 · answered by Anonymous · 0⤊ 2⤋
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2014-08-15 01:31:20 · answer #5 · answered by Anonymous · 0⤊ 2⤋
Check out this site.
http://archives.math.utk.edu/visual.calculus/3/applications.4/index.html
I hope it helps more than it confuses (haven't had a chance to test out the formula in the example)
2006-11-01 04:04:32 · answer #6 · answered by QnA 2 · 0⤊ 4⤋