Chapter 10.3, Problem 96E

Tractrix A person moves from the origin along the positive y-axis pulling a weight at the end of a 12-meter rope. Initially, the weight is located at the point (12.0).

(a) In Exercise 61 of Section 8,4, it was shown that the path of the weight is modeled by the rectangular equation

$y=-12\ln \frac{12-\sqrt{144-x^2}}{x}-\sqrt{144-x^2}$

where $0<x\le 12$. Use a graphing utility to graph the rectangular equation.

(b) Use a graphing utility to graph the parametric equations

$y=12\text{sech}\frac{t}{12}\text{and}y=t-12\tanh \frac{t}{12}$

where $t\le 0$. How does this graph compare with the graph in pan (a)? Which graph (if either) do you think is a better representation of the path?

(c) Use the parametric equations for the tractrix to verify that the distance from the v-intercept of the tangent line to the point of tangency is independent of the location of the point of tangency.