Question

-/1 points a ladder 12 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 resources read it watch it master it submit answer 13. -/1 points find the indicated derivative by finding the first - few derivatives and recognizing the pattern that occurs. $\frac{d^{93}}{dx^{93}}(cos(x))$

Explanation:

Step1: Establish the relationship

We know that $\sin\theta=\frac{x}{12}$, so $x = 12\sin\theta$.

Step2: Differentiate with respect to $\theta$

Using the derivative formula $\frac{d}{d\theta}(\sin\theta)=\cos\theta$, we get $\frac{dx}{d\theta}=12\cos\theta$.

Step3: Evaluate at $\theta=\frac{\pi}{3}$

Substitute $\theta = \frac{\pi}{3}$ into $\frac{dx}{d\theta}$. Since $\cos\frac{\pi}{3}=\frac{1}{2}$, then $\frac{dx}{d\theta}\big|_{\theta=\frac{\pi}{3}}=12\times\frac{1}{2}=6$.

for the second - part:

Step1: Find the first few derivatives of $y = \cos(x)$

The first derivative: $y'=-\sin(x)$; the second derivative: $y''=-\cos(x)$; the third derivative: $y'''=\sin(x)$; the fourth derivative: $y^{(4)}=\cos(x)$.

Step2: Identify the pattern

The derivatives of $\cos(x)$ repeat every 4 derivatives. We can write the general formula for the $n$-th derivative of $\cos(x)$ as $y^{(n)}=\cos(x + \frac{n\pi}{2})$.

Step3: Calculate the 93 - rd derivative

Divide 93 by 4: $93 = 4\times23+1$. So $n = 93$ gives $y^{(93)}=\cos(x+\frac{93\pi}{2})=\cos(x + 46\pi+\frac{\pi}{2})$. Since $\cos(A + 2k\pi)=\cos(A)$ for integer $k$, then $y^{(93)}=-\sin(x)$.

Answer:

$6$