Chapter 3.4, Problem 9ES

Let $\theta$ (in radians) be an acute angle in a right triangle, and let $x$ and $y$ , respectively, be the lengths of the sides adjacent to and opposite $\theta$ . Suppose also that $x$ and $y$ vary with time.

(a) How are $d\theta /dt,dx/dt$ , and $dy/dt$ related?

(b) At a certain instant, $x=2$ units and is increasing at $1\text{ unit/s}$ , while $y=2$ units and is decreasing at $\frac{1}{4}\text{ unit/s}$ . How fast is $\theta$ changing at that instant? Is $\theta$ increasing or decreasing at that instant?