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Leaning Ladders |
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Consider two ladders leaning in opposite directions between two parallel walls. One ladder is 40 feet long and the other is 60 feet long. Each ladder is touching the base of the wall opposite the wall on which it is leaning. The two ladders cross at a point 10 feet above the level ground. How far apart are the walls? |
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This type of problem often leads to questions about what constitutes a "closed-form" solution, and about the relative utility of implicit vs explicit definitions. If the tops of the ladders contact the walls at elevations a,b, then the height h of the crossing point is given by |
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regardless of the distance between the two walls. Of course, if w is the distance between the walls and A,B are the lengths of the two ladders, then |
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Thus, given the value of w we can directly compute the corresponding value of h. Notice, however, that this computation involves the evaluation of square roots, which is an algorithmic process that generally yields only approximate answers. |
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Now suppose, instead, that we're given some value of h and asked to compute the value of w. Our first impulse is to seek a formula involving only standard "calculator key" operations giving w as an explicit function of h. To do this we would probably try to “invert" equation (1) and isolate w. If we define D = A2 - B2 and x = B2 - w2, this leads to the quartic |
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For example, given the values A = 40, B = 60, h = 10, and making the substitution x = 200y, we arrive at |
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Taking the appropriate root y = 0.81149... of this equation, and noting that w is the square root of 1600 - 200y, we have w = 37.917 feet. This would typically be presented as "the solution" of the problem. However, notice that (2) is no more explicit than (1). Rather than deriving the polynomial (2) and then solving for the root, we could just as easily have immediately solved for the root of the equation |
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since most root-finding algorithms will work just as well with a general function. The only real advantage in converting to a pure polynomial is that polynomials are sort of a "canonical form" and we can check easily for possible factorizations, etc. If, for example, we find that the polynomial form is irreducible, then we feel confident we're not missing some trivial simplification, which may not be obvious from equation (3). |
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