How long is the ladder? Torsten Sillke, July 1998
Two ladders of equal length are hinged at one end and
stand as an inverted V, with their bottem ends 4m apart.
When a person climbs 3m up one the ladders, the rung he
stands on is as far from the top of the ladder as it is
from the bottom of the opposite ladder.
How long are the ladders?
V
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ P
/ . \
/ . \
A ___________________ B
- Brian Bolt;
A mathematical Pandora's Box,
Cambridge Univ. Press, 1993.
Problem 116: How long is the ladder?
Solution:
AV = BV = 2 ( 1 + sqrt(7) ) meter = 7.2915 meter
Trigonometric Solution:
Scale the figure so that AP = PV = 1.
Let the angle(AVB) = 2v then we get
AV = 2 cos(2v)
BP = BV - VP = 2 cos(2v) - 1
AB = 2 AV sin(v) = 4 sin(v) cos(2v).
V
/ \
/ \
/ 2 v \
2 cos(2v) / \ 1
/ \
/ \
/ \
/ P
/ . \ 2 cos(2v) - 1
/ . \
A ___________________ B
4 sin(v) cos(2v)
Substitute cos(2v) = 1 - 2 sin^2(v) according to
the addition theorem and letting s = 2 sin(v) we get
AV = 2 - s^2
BP = 1 - s^2
AB = 2s - s^3 = s AV
Given BP and AB we have with r := AB/BP the relation:
2s - s^3 1
r = --------- = s - -------
1 - s^2 s - 1/s
This gives the cubic equation in s
s^3 - r s^2 - 2 s + r = 0
The desired result is AV = AB/s.
Nice Parameters (cubic has a rational root):
Let w = AB, a = BP, and d = AV.
w = 3/2 a = 5 d = 3/2 (3 + sqrt(7)) = 10.112486
w =sqrt(3) a = 2 d = 1/2 (3 + sqrt(11)) = 4.372281
w = 2 a = 3/2 d = 1 + sqrt(7) = 3.645751
w = 5/2 a =21/17 d = 5/34 (5 + sqrt(382)) = 3.609532
w = 3 a = 8/7 d = 3/14 (3 + sqrt(233)) = 3.913786
w = 4 a =15/14 d = 2/7 (2 + sqrt(214)) = 4.751068
Finding nice parameters:
The parameter (a,w,d) must satisfy the cubic
d^3 - 2a d^2 - w^2 d + a w^2 = 0
Isolating a we get
a/d = (d^2 - w^2)/(2d^2 - w^2)
where 0 <= w <= d. If we select rational w and d we will
get a rational parameter a.
Set d=1 then a = (1-w^2)/(2-w^2) but now let w^2 > 2.
Then we have selected the wrong root d=1.
Now the cubic factors as d=1 is a root per construction into
(d - 1)( d^2 - w^2/(w^2 - 2) d - w^2(w^2 - 1)/(w^2 - 2) ) = 0
As d>=a the wanted root is the positive root of the quadratic factor
d = 1/2 w/(w^2-2) (w + sqrt(4(w^2-1)(w^2-2)+w^2))
--
mailto:Torsten.Sillke@uni-bielefeld.de
http://www.mathematuk.uni-bielefeld.de/~sillke/