On Sun, 29 May 2005 17:22:04 GMT, machineman
wrote:
That jives with my cad drawing answer. Imaginary numbers give me a
headache :-)
Marvin W. Klotz wrote:
On Sun, 29 May 2005 01:05:13 GMT, Errol Groff wrote:
Ladder_Problem.PDF is an interesting math problem I found in the book
Machine Shop Trade Secrets by James A Harvey.
The solution I used to find the answer will be posted as
Ladder_Problem_Answer.PDF
http://metalworking.com/DropBox/
Errol Groff
Instructor, Machine Tool Department
H.H. Ellis Technical High School
643 Upper Maple Street
Dantieson, CT 06239
New England Model Engineering Society
www.neme-s.org
This isn't a trig problem. Rather it's an algebra problem that, surprisingly,
requires the solution to a quartic equation. However, by some clever
selection of variables, the quartic can be separated into two quadratics.
b = side of box
l = length of ladder
y = height of smallest triangle
x = base of medium sized triangle
Then the base of the large triangle is x+b and its height is y+b.
From similar triangles, we have:
y/b = b/x = x*y = b^2 = y = b^2/x (1)
Applying Pythagoras to the large triangle:
(x+b)^2 + (y+b)^2 = l^2
or:
x^2 + 2*b*x + b^2 + y^2 + 2*b*y + b^2 = l^2
Substituting y = b^2/x from (1) yields:
x^2 + 2*b*x + b^2 + b^2/x^2 + 2*b*b^2/x + b^2 = l^2 (2)
Now define:
a = x + b^2/x (3)
so:
a^2 = x^2 + 2*b^2 + b^4/x^2
Then (2) becomes:
a^2 + 2*b*a - l^2 = 0
Solve this quadratic for a.
Now substitute a into (3) and solve the resulting quadratic to find x.
a = x + b^2/x (3)
x^2 - a*x + b^2 = 0
Grinding through the numbers (I used a program I wrote):
Length of ladder [25] ? 120
Side of box [6] ? 24
Solutions to:
+1.0000 * x^2 +48.0000 * x^1 -14400.0000 = 0
a
real: 98.376468 imaginary: 0.000000
real: -146.376468 imaginary: 0.000000
a selected = 98.3765
Solutions to:
+1.0000 * x^2 -98.3765 * x^1 +576.0000 = 0
a
real: 92.124028 imaginary: 0.000000
real: 6.252440 imaginary: 0.000000
Base and height of large triangle a
116.1240, 30.2524 or 30.2524, 116.1240
So the value labeled X in the PDF is 116.124
Regards, Marv
Home Shop Freeware - Tools for People Who Build Things
http://www.geocities.com/mklotz.geo
True, CAD will get you the answer but it's just so... inelegant.
There's a big difference between getting the answer and solving the problem.
Kind of like the difference between buying and making a needed part.
Regards, Marv