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(CHAIN RULE APPLICATION) How fast is the beam of light moving along the shoreline when it is 1 km from P?
A lighthouse is located on a small island, 3 km 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 1 km from P?
Let d be the distance from P
and θ be the angle the light makes
tanθ = x/3
sec²θ dθ/dt = 1/3 * dx/dt
dθ/dt = 4 revolutions per minute
= 8π radians/min
When x = 1
tanθ = 1/3
θ = arctan(1/3)
sec²θ dθ/dt = 1/3 * dx/dt
sec²(arctan(1/3)) 8π = 1/3 * dx/dt
(1 – tan²(arctan(1/3)) 24π = dx/dt
dx/dt = 24π (1 – 1/9)
= 64π/3 km/min
Strange Beam of Light Wiltshire
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