Single Variable Calculus, Chapter 3, 3.4, Section 3.4, Problem 37

Suppose that a ladder 10$ft$ long nests against a vertical wall. Let $\theta$ be the angle be the angle between the top of the ladder and the wall and let $x$ be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does $x$ change with respect to $\theta$ when $\displaystyle \theta = \frac{\pi}{3}$




We can use the sine function to group all t the given quantity in one equation thus,

$\displaystyle \sin \theta = \frac{x}{10}$
$x = 10 \sin \theta$


$\begin{equation} \begin{aligned} \frac{d}{d\theta} &= 10 \sin \theta = 10 \frac{d}{dx} \sin \theta\\ \\ \frac{d}{d\theta} &= 10 \cos \theta ;\text{ when } \theta = \frac{\pi}{3}\\ \\ \frac{d}{d\theta} &= 10 \cos \left( \frac{\pi}{3}\right)\\ \\ \frac{d}{d\theta} &= 5 \frac{ft}{rad} \end{aligned} \end{equation}$