a lighthouse stands 550 m off a straight shore and the focused beam of its light revolves (at a constant…

Explanation:

Step1: Find the angular velocity

The light revolves (4) times per minute. One full revolution is (2\pi) radians. So the angular velocity (\omega) is (\omega = 4\times2\pi) radians per minute. (\omega=8\pi) radians per minute.

Step2: Establish the relationship between (x) and (\theta)

Let (x) be the distance from (P) to the point where the beam hits the shore and (\theta) be the angle between the line from the lighthouse to (P) and the line from the lighthouse to the point on the shore. We know that (\tan\theta=\frac{x}{550}), so (x = 550\tan\theta).

Step3: Differentiate (x) with respect to (t)

Using the chain - rule, (\frac{dx}{dt}=550\sec^{2}\theta\frac{d\theta}{dt}). Since (\frac{d\theta}{dt}=\omega = 8\pi) and when (x = 275), (\tan\theta=\frac{275}{550}=\frac{1}{2}). Then (\sec^{2}\theta=1 + \tan^{2}\theta=1+\left(\frac{1}{2}\right)^{2}=\frac{5}{4}).

Step4: Calculate (\frac{dx}{dt})

Substitute (\sec^{2}\theta=\frac{5}{4}) and (\frac{d\theta}{dt}=8\pi) into (\frac{dx}{dt}=550\sec^{2}\theta\frac{d\theta}{dt}). (\frac{dx}{dt}=550\times\frac{5}{4}\times8\pi) [ \begin{align*} \frac{dx}{dt}&=550\times10\pi\ &=5500\pi \end{align*} ]

Answer:

(17279) m/min

For the variation: Since (\frac{dx}{dt}=550\sec^{2}\theta\frac{d\theta}{dt}) and (\sec^{2}\theta = 1+\tan^{2}\theta=1+\left(\frac{x}{550}\right)^{2}), as (x) (the distance between (P) and (Q)) increases, (\frac{dx}{dt}) (the speed of the beam along the shore) increases quadratically.