Crossed Ladders Problem

Crossed Ladders Problem

File:20191011crossedLadders00.png

Figure 1.

The Crossed ladders Problem.
$w=w_1+w_2.$

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. Template:RoundBoxBottom

Solution

File:20191011crossedLadders01.png

Figure 2.

$f(A),g(B)$ touch where $A=B=h.$ ${\displaystyle }$ ${\displaystyle }$

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 $f^{\prime }(A)=g^{\prime }(B)=-2h^3.$

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

Calculating A₀, A₁

File:20191011crossedLadders02.png

Figure 3.

Calculation of $A_0,A_1$

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

File:20191011crossedLadders03.png

Figure 4.

Calculation $B_0,B_1.$

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

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

File:20191011crossedLadders04.png

Figure 5.

Interpretation of $A_1,B_1$

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 $h=240.$ Template:RoundBoxBottom