Question

a ladder 20 ft long rests against a vertical wall. let $\theta$ 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 (in ft/rad) does $x$ change with respect to $\theta$ when $\theta=\frac{pi}{3}$? ft/rad

Explanation:

Step1: Establish trigonometric relation

We know that $\sin\theta=\frac{x}{20}$ (since in a right - triangle formed by the ladder, wall and ground, $\sin\theta$ is the ratio of the opposite side $x$ to the hypotenuse 20). So, $x = 20\sin\theta$.

Step2: Differentiate with respect to $\theta$

Differentiating $x$ with respect to $\theta$ using the derivative of the sine function. $\frac{dx}{d\theta}=20\cos\theta$.

Step3: Substitute $\theta=\frac{\pi}{3}$

Substitute $\theta = \frac{\pi}{3}$ into $\frac{dx}{d\theta}$. We know that $\cos\frac{\pi}{3}=\frac{1}{2}$. So, $\frac{dx}{d\theta}\big|_{\theta=\frac{\pi}{3}}=20\times\frac{1}{2}=10$.

Answer:

$10$