Question

10 a lighthouse is on an island 1 mile off the coast. its light rotates at 10 revolutions per minute. to an observer 2 miles away from the closest point on the shore, how fast does the beam appear to be moving along the coast? assume the coast is a straight line.

Explanation:

Step1: Convert angular - speed to radians per minute

One revolution is $2\pi$ radians. If the light rotates at 10 revolutions per minute, then the angular - speed $\omega$ is $\omega = 10\times2\pi=20\pi$ radians per minute.

Step2: Set up a trigonometric relationship

Let $x$ be the distance along the coast from the point on the shore closest to the lighthouse to the point where the beam hits the shore, and $\theta$ be the angle between the line from the lighthouse to the closest point on the shore and the line from the lighthouse to the point where the beam hits the shore. We know that $\tan\theta=\frac{x}{1}$ (since the distance from the lighthouse to the shore is 1 mile), so $x = \tan\theta$.

Step3: Differentiate with respect to time

Differentiate both sides of $x=\tan\theta$ with respect to time $t$. Using the chain - rule, $\frac{dx}{dt}=\sec^{2}\theta\frac{d\theta}{dt}$.

Step4: Find the value of $\theta$ for the given observer's position

The observer is 2 miles away from the closest point on the shore. So when $x = 2$, $\tan\theta=2$. Then $\sec^{2}\theta=1 + \tan^{2}\theta=1 + 4=5$.

Step5: Substitute the values of $\sec^{2}\theta$ and $\frac{d\theta}{dt}$

We know that $\frac{d\theta}{dt}=20\pi$ radians per minute and $\sec^{2}\theta = 5$. Substituting these values into $\frac{dx}{dt}=\sec^{2}\theta\frac{d\theta}{dt}$, we get $\frac{dx}{dt}=5\times20\pi = 100\pi$ miles per minute.

Answer:

$100\pi$ miles per minute