A lighthouse is located 3km away from the nearest point P on a straight shoreline
and its light makes four revolutions per minute.
How fast is the beam of light moving along the shoreline when it is 1km from P?
| θ *
3 | *
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P x B
The lighthouse is at L.
Its beam shine at point B on the shore, x km from point P.
Let θ = angle BLP.
The light makes 4 revolutions per minute.
Each revolution is 2π radians.
Hence, dθ/dt = 8π radians/minute.
We have: .tanθ .= .x/3 . → . x .= .3·tanθ
Differentiate with respect to time: .dx/dt .= .3·sec˛θ·(dθ/dt) .
When x = 1, we have a right triangle with angle θ.
The opposite side is 1, the adjacent side is 3.
. . Then the hypotenuse is √10.
Hence: .secθ = √10/3 .and .sec˛θ = 10/9
Substitute into : .dx/dt .= .3(10/9)(8π) .= .80π/3 km/min