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 rotation rate to radians per minute

The light rotates at 10 revolutions per minute. Since 1 revolution = $2\pi$ radians, the angular - velocity $\omega=10\times2\pi = 20\pi$ 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 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 observer. We have $\tan\theta=\frac{x}{1}$ (where 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 $\sec^{2}\theta$ for the given situation

The observer is 2 miles away from the closest point on the shore. So, $\tan\theta = 2$ (since $x = 2$ and $x=\tan\theta$). Using the identity $\sec^{2}\theta=1 + \tan^{2}\theta$, we get $\sec^{2}\theta=1 + 4=5$.

Step5: Substitute values to find $\frac{dx}{dt}$

We know that $\frac{d\theta}{dt}=\omega = 20\pi$ 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