Asked by Noah
A lighthouse stands 450 m off a straight shore and the focused beam of its light revolves four 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 250 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.
Answers
Answered by
oobleck
Let x be the distance PQ. We want to find how fast Q is moving -- that is, dx/dt
If you draw the diagram, you can see that when the angle θ = PLQ,
tanθ = x/450
so
sec^2θ dθ/dt = 1/450 dx/dt
and since the light rotates 4 times per minute, one sweep takes 15 seconds. So dθ/dt = 2π/15 rad/s
Now plug all that in to find dx/dt when x=250
tanθ = 250/450 = 5/9
so sec^2θ = 1 + 25/81 = 106/81
106/81 * 2π/15 = 1/450 dx/dt
x = 2120π/27 ≈ 246.67 m/s
If you draw the diagram, you can see that when the angle θ = PLQ,
tanθ = x/450
so
sec^2θ dθ/dt = 1/450 dx/dt
and since the light rotates 4 times per minute, one sweep takes 15 seconds. So dθ/dt = 2π/15 rad/s
Now plug all that in to find dx/dt when x=250
tanθ = 250/450 = 5/9
so sec^2θ = 1 + 25/81 = 106/81
106/81 * 2π/15 = 1/450 dx/dt
x = 2120π/27 ≈ 246.67 m/s
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