A lighthouse is 100 meters from a straight shoreline.

Question

a lighthouse is 100 meters from a straight shoreline. the light turns at a rate of 10 revolution per minute (20π radians/minute), and shines a moving spot of light along the shore. How fast is the spot of light moving when it's 100 meters from the point on the shore which is nearest to the light house? be sure to include the correct units in your answer.

Answer 1

Let r=length of light beam from lighthouse to spot  on shore.

     θ=angle from perpendicular to shore from lighthouse to beam.

    100/r = =cosθ  or r=100secθ

dr/dt = 100 secθ tanθ dθ/dt, which for θ=π/4, the right triangle is formed by 100 m from the beach to lighthouse, 100 m along the beach, to the spot, and the beam.

dr/dt = 100√2×1×20π = 2000√2π=8885.765876...m/min = 148.0960979 m/sec

Answer 2

linear velocity = angular velocity x radius

 

radius = 100√2 meters

 

angular velocity = 20π radians/minute

 

linear velocity = 20π rad/min x (100√2 meters) = 8881 meters / min