A lighthouse is located on a small island 5 km away from the nearest point P on a straight shoreline and its light makes seven revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?

The movement of the beam of light is an illustration of distance and rates

The beam of light along the shoreline is moving at 228.8 km per minute when it is 1 km from P

From the question, we have:

Convert to radians per mins

Take tan of

Differentiate both sides with respect to t

Multiply both sides by 5

Substitute

Next, calculate the hypotenuse (h) side of the triangle, using the following Pythagoras theorem

From the question, x = 1.

So, we have:

Take square roots

Calculate cos theta

Take inverse, to calculate secant

Substitute in

Rewrite as:

So, we have:

Hence, the beam of light along the shoreline is moving at 228.8 km per minute when it is 1 km from P

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Light is moving at the speed = 77.78 km/min

Step-by-step explanation:

Given:

Distance between island and point P = 5 km

Moving along the the shoreline when it is 1 km from P.

So, x = 1

From the above statement light makes seven revolutions per minute.

Therefore,              

Solution:

From the given figure.

Substitute x = 1

 -------------------(1)

Now, first we differentiate both side with respect to t.

Using chain rule.

Substitute from equation 1.

Substitute

Applying cross multiplication rule.

Therefore, light is moving along the shore at the speed of 77.78 km/min

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