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      <text>2.4.6 (Application - Crossed Ladders Problem)</text>
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      <text><![CDATA[<h1>Crossed Ladders Problem</h1>
<p>We have already seen that cubic equations can arise when we calculate volumes, and therefore in order to understand volumes, we need to understand cubics. Quartic polynomials also have a variety of applications across both mathematics and physics. For example, in manufacturing, a torus can be created using an endmill cutter and to find the general position of a torus relative to a triangulated surface requires solving a quartic. Also, quartics can arise in the field of optics. One such application is Alhazen's problem which asks '<em>if light emanates from one point and reflects from a spherical mirror to a second given point, at which point does the light ray strike the mirror?</em>', it transpires that in order to solve this, we must solve a quartic equation. The application we are going to examine here is often referred to as the 'Crossed ladders problem.' It is mainly raised by mathematicians who like the problem as it is relatively easy to state but leads to more complicated mathematics than one would initially expect. However, it does represent some interesting geometrical properties.</p>
<p> </p>
<h3>The Problem</h3>
<p>Suppose that two ladders lie oppositely across an alleyway, of lengths \(a\) and \(b\) respectively. Suppose the ladders cross at a height of \(h\) above the floor of the alley. What is the width \(w\) of the alley?</p>
<h3>The Solution</h3>
<p>We can model the set up described in the problem with the following diagram.</p>
<p> </p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Crossed%20Ladders.png" alt="Diagram illustrating the crossed ladders problem" width="487" height="338" /></p>
<p> </p>
<p>To solve this problem begin by letting \(A\) and \(B\) represent the heights of the walls where the ladders lean. We can divide the width of the alley into two parts - either side of the point at which it meets \(h\). Then we proceed by looking at the similar triangles illustrated below. </p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Crossed%20Ladders%202.png?time=1664786422334" alt="Imagine showing diagram of crossed ladders problem" width="483" height="325" /></p>
<p>We can see here that the blue and red triangles are similar, therefore their side lengths are in proportion. We can create a second set of similar triangles by considering the triangles with heights, \(h\) and \(A\) therefore we have \[ \frac{B}{w} = \frac{h}{w_1} \qquad \mbox{ and } \qquad \frac{A}{w} = \frac{h}{w_2}, \] where \(w_1, w_2 &gt; 0\) (otherwise it would make no sense to divide by them). Now we can rearrange to obtain \[ \frac{B}{w} = \frac{h}{w_1} \iff w_1 = \frac{h}{B}w \qquad \mbox{and} \qquad \frac{A}{w} = \frac{h}{w_2} \iff w_2 = \frac{h}{A}w. \] Hence we have \[ \begin{align*} w &amp;= w_1 + w_2 \\ &amp;= \frac{h}{B}w + \frac{h}{A}w \\ &amp;= w (\frac{h}{A} + \frac{h}{B}). \end{align*} \] This holds if and only if \(1 = \frac{h}{A} + \frac{h}{B}\) or in other words if \[ \frac{1}{h} = \frac{1}{A} + \frac{1}{B}. \]</p>
<p>The next step is to rewrite the values of \(A\) and \(B\) by using Pythagoras' theorem on each of the similar triangles. Thus \[ a^2 = A^2 + w^2 \qquad \mbox{and} \qquad b^2 = B^2 + w^2. \] We can now subtract the second of these equations from the first to get \[ a^2-b^2 = A^2-B^2 = (A+B)(A-B). \] We can then combine this with the fact that \(\frac{1}{h} = \frac{1}{A} + \frac{1}{B},\) and solving for \(A\) to give \[ A^4 - 2hA^3 + (h-A)^2(a^2-b^2) = 0. \] Alternatively we could solve for \(B\) instead to obtain \[ B^4 -2hB^3 + (h-B)^2(b^2-a^2) = 0. \]</p>
<p>In either case we can see that we have a quartic in the leaning heights of each ladder. Therefore if we wanted to find the width of the alleyway we could solve these quartic for either \(A\) or \(B\) to obtain the leaning height of the ladder, then we could use Pythagoras' Theorem to obtain the required width \(w\).</p>]]></text>
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