Crossed Ladders Problem

=Crossed Ladders Problem=



The Crossed ladders problem is usually classified as mathematical entertainment. However, it leads to some pretty curves, the solutions of two quartic equations and the theoretical interpretation of unanticipated results.

The figure shows two ladders across an alley. Ladder $$a$$ has its feet on the ground against wall $$A$$ and is leaning with its top against wall $$B.$$

Ladder $$b$$ has its feet on the ground against wall $$B$$ and is leaning with its top against wall $$A.$$

$$h$$ is the height above ground at which the ladders cross.

Three known values are the lengths $$a,\ b,\ h.$$ What is the width $$(w = w_1 + w_2)$$ of the alley?

In this example the known values are: $$a,\ b,\ h = 525,\ 1092,\ 240$$ units.

Solution


The relevant functions are:

$$f(A) = A^4 - 480A^3 - 916839A^2 + 440082720A - 52809926400$$

$$g(B) = B^4 - 480B^3 + 916839B^2 - 440082720B + 52809926400$$

When $$A = B = h,\ f(A) = g(B) = -h^4$$ and $$f'(A) = g'(B) = -2h^3.$$

The figure shows that the curves touch where $$x = h = 240.$$

Calculating A0, A1


Length $$A$$ is a little less than length $$b = 1092.$$ We use $$x = 1092$$ as the starting point and Newton's method quickly finds $$A_0 = 1008.$$ With $$A, b$$ known $$w = \sqrt{ b^2 - A^2 } = 420.$$

The following calculations are for the interpretation of the values $$A_1, B_1.$$

Take the known value out of $$f(A)$$ and the remaining cubic function is:

$$x^3 + 528x^2 - 384615x + 52390800.$$ The one real root of this function is: $$A_1 = -976.706982646915.$$

Calculating B0, B1


$$B = \sqrt{a^2 - w^2} = 315.$$

Take the known value out of $$g(B)$$ and the remaining cubic function is:

$$x^3 - 165x^2 + 864864x -167650560.$$ The one real root of this function is: $$B_1 = 192.659102954523.$$

Interpretation of A1, B1


Ladder $$a$$ has its feet at point $$(-w_1, 0)$$ and it is at rest against point $$(w_2, B_1 = 192.659102954523).$$

Ladder $$b$$ has its feet at point $$(-w_1, A_1 = -976.706982646915 )$$ and it is at rest against point $$(w_2, 0).$$

Both ladders extended intersect at height $$h = 240.$$