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 2 km away from the nearest point P on a straight shoreline and its light makes 5 revolutions per minute. (Round your answer to one decimal place.) ____________________km/min.  

Expert Answers

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You should sketch a triangle MPN, where MP represents the horizontal distance of 2 km between lighthouse and the point P and NP represents the beam of light.

Evaluating the tangent of angle `hat(NPM)`  yields:

`tan hat(NPM) = (NM)/(MP) => tan hat(NPM) = (NM)/2`

You may use the following notation for the angle `hat(NPM)`  such that:

`hat(NPM) = theta`

The problem provides the information that `(d theta)/(dt) = 5`  rev/min

Differentiating `tantheta = (NM)/2`  with respect to time t yields:

`sec^2 theta ((d theta))/(dt) = (1/2) (d(NM))/dt`

You shold evaluate theta for NM = 1 Km such that:

`tan theta = 1/2 => theta = arctan(1/2)`

`sec^2(arctan(1/2))*5 =(1/2) (d(NM))/dt`

You need to evaluate how fast the beam of light moving along the shoreline when it is 1 km from P such that:

`(d(NM))/dt = 10sec^2(arctan(1/2))`

Hence, evaluating how fast the beam of light moving along the shoreline when it is 1 km from P yields `(d(NM))/dt = 10sec^2(arctan(1/2)).`

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