A lighthouse stands 525 m off a straight shore and the focused beam of its light

Question

A lighthouse stands 525 m off a straight shore and the focused beam of its light revolves two times each minute. As shown in the figure, P is the point on shore closest to the lighthouse and Q is a point on the shore 300 m from P. What is the speed of the beam along the shore when it strikes the point Q? Describe how the speed of the beam along the shore varies with the distance between P and Q. Neglect the height of the lighthouse. 525 m 300 m Let x be the distance along the shore from point P to the beam and let be the angle through which the light revolves. Write an equation that relates x and a Differentiate both sides of the equation with respect to t. dx dt (D dt When the beam strikes the point Q, its speed along the shore is about (Do not round until the final answer. Then round to the nearest integer as needed.) How does the speed of the beam along the shore vary with the distance between P and Q? O A. If Q were closer to P, then the speed of the beam along the shore when it strike Q would be greater. O B. The speed of the beam along the shore is a constant because it depends only on the rate at which the light revolves. O C. The speed of the beam along the shore is a constant because it depends only on the distance from the lighthouse to P Click to select your answer(s).

Explanation / Answer

equation is tan =x/525

=>x=525tan

=>dx/dt =(525sec2) (d/dt)

at Q , x=300

tan=300/525

=>=0.519146 radians

light rotates 2 times each minute constantly

=>d/dt=(2*2/60)

=>d/dt=(/15) radians per second

dx/dt =(525sec2(0.519146))(/15)

=>dx/dt =145.8596589

when beam strikes the point Q ,its speed along the shore is about 146 meters/second

speed of the beam along the shore at point Q increases with the increase in distance between P,Q

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