4. a revolving lighthouse is 500 m across from a straight shore. the beam of light is traveling 240 m/s at a…

Explanation:

Step1: Establish a relationship

Let $x$ be the distance along the shoreline from the point on the shore closest to the lighthouse, and $y = 500$m be the distance from the lighthouse to the shore. Let $\theta$ be the angle between the line - from the lighthouse to the point on the shore closest to it and the line - from the lighthouse to the point on the shore where the beam of light hits. We know that $\tan\theta=\frac{x}{y}$. Since $y = 500$, we have $\tan\theta=\frac{x}{500}$.

Step2: Differentiate both sides with respect to time $t$

Using the chain - rule, $\sec^{2}\theta\frac{d\theta}{dt}=\frac{1}{500}\frac{dx}{dt}$.

Step3: Find $\sec^{2}\theta$ when $x = 600$

First, when $x = 600$ and $y = 500$, by the Pythagorean theorem, the distance from the lighthouse to the point on the shore where the beam hits is $r=\sqrt{500^{2}+600^{2}}=\sqrt{250000 + 360000}=\sqrt{610000}$. And $\sec\theta=\frac{\sqrt{x^{2}+y^{2}}}{y}$. When $x = 600$ and $y = 500$, $\sec\theta=\frac{\sqrt{600^{2}+500^{2}}}{500}=\frac{\sqrt{360000 + 250000}}{500}=\frac{\sqrt{610000}}{500}$. So, $\sec^{2}\theta=\frac{610000}{250000}=\frac{61}{25}$.

Step4: Substitute known values

We know that $\frac{dx}{dt}=-240$ m/s (negative because $x$ is changing with respect to time). Substitute $\sec^{2}\theta=\frac{61}{25}$ and $\frac{dx}{dt}=-240$ into $\sec^{2}\theta\frac{d\theta}{dt}=\frac{1}{500}\frac{dx}{dt}$. We get $\frac{61}{25}\frac{d\theta}{dt}=\frac{1}{500}\times(-240)$.

Step5: Solve for $\frac{d\theta}{dt}$

First, simplify the right - hand side: $\frac{1}{500}\times(-240)=-\frac{12}{25}$. Then, solve for $\frac{d\theta}{dt}$: $\frac{d\theta}{dt}=-\frac{12}{25}\times\frac{25}{61}=-\frac{12}{61}$ rad/s. The negative sign just indicates the direction of rotation. The magnitude of the angular velocity is $\frac{12}{61}$ rad/s.

Answer:

$\frac{12}{61}$ rad/s