Parametric equations : an investigation into ladder applications

Parametric equations are used to represent the pathway of an object in terms of time or another changing variable. This allows, for example, for equations that are written using two variables to be examined in terms of the passage of time. In this paper the author examines two traditional application problems whose solutions can be enriched through the use of parametric equations. In the first, the falling ladder problem, a ladder is leaned against a wall then pulled away with a constant velocity. Deriving parametric equations for this scenario permits the pathway of the ladder to be plotted. Parametric equations also make it possible for the horizontal and vertical velocities of the ladder to be examined separately. The second problem is that of maximizing the length of a ladder that can fit around a hallway corner. In this problem an envelope algorithm is first developed, then parametrized to further investigate this scenario. Using these two situations, this report ultimately shows how parametric equations can be used to give a more thorough approach to some of today’s most classic calculus problems.