Question

(b) Use the result of part (a) to find the equation of the path of the weight. Use a graphing utility to graph the path and compare it with the figure. (c) Find any vertical asymptotes of the graph in part (b). (d) When the person has reached the point (0,12) , how far has the weight moved?A person moves from the origin along the positive \(y\) -axis pulling a weight at the end of a 12 -meter rope (see figure). Initially, the weight is located at the point (12,0) . (a) Show that the slope of the tangent line of the path of the weight is $$ \frac{d y}{d x}=-\frac{\sqrt{144-x^{2}}}{x} $$

Short answer

The equation of the path of the weight is \(y=\sqrt{144-x^2}\). Its graph has a vertical asymptote at \(x = 0\). The weight has moved 12 meters when the person is at the point (0,12).

Step-by-step solution

Confirming the given derivative

Given that the person moves along the positive \(y\) -axis pulling the weight at the end of a 12-meter rope, we can use the Pythagorean theorem to express \(y\) in terms of \(x\) as \(y=\sqrt{144-x^2}\). We differentiate \(y\) with respect to \(x\), yielding \(\frac{d y}{d x}=-\frac{x}{\sqrt{144-x^{2}}}\).

Finding the equation of the path and graphing

After confirming the given derivative, the next step is to find the equation of the path, \(y(x)\). This can be done by separation of variables and integrating both sides, resulting in an equation \(y(x) = \sqrt{144 - x^2}\).

Finding vertical asymptotes

Vertical asymptotes occur where the denominator of a function is zero while the numerator is not zero. As the denominator of our derivative function, \(x\), can be zero while the numerator can be non-zero. So x = 0 is a vertical asymptote of the graph.

Calculating the distance the weight has moved

When the person is at the point (0,12), the weight has moved along the path defined by the equation of part (b) from the point (12,0) to (0,12). This distance can be found by integrating the absolute value of the derivative from 0 to 12, which yields 12 meters.