The beacon on a lighthouse makes one revolution every 20 seconds. The beacon is 300 feet from

the nearest point, P, on a straight shoreline. Find the rate at which the ray of light moves along
the shore at a point 200 feet from P.

Make a sketch, let the distance between the end of the ray of light and point P be x ft

let the angle formed at that moment be Ø
then,
tanØ = x/300
x = 300tanØ
dx/dt =300sec^2 Ø dØ/dt

let the hypotenuse be h
when x=200
h^2 = 200^2+300^2
h = 100√13
secØ = 100√13/300 = √13/3
sec^2 Ø = 13/9
and we are told dØ/dt = 2π/20 rad/sec
= π/10 rad/sec

dx/dt = 300(13/9)(π/10) = 130π/3 ft/sec

To find the rate at which the ray of light moves along the shore at a point 200 feet from point P, we can use the concept of similar triangles.

Let's denote the distance from point P to the point on the shore where the ray of light is located as x. We know that this distance is 200 feet.

We can create a right triangle with the following sides:
1. The hypotenuse, representing the distance from the lighthouse beacon to the point on the shore, which is 300 feet.
2. The side adjacent to the angle between the hypotenuse and the shoreline, which is x.
3. The side opposite to the angle between the hypotenuse and the shoreline, which represents the distance at which the ray of light moves along the shore.

Since the lighthouse beacon makes one revolution every 20 seconds, we know that it completes a full circle in 20 seconds. Therefore, the angle between the hypotenuse and the shoreline changes at a constant rate.

We can now solve the problem using similar triangles. The rate at which the ray of light moves along the shore is equal to the rate at which the angle changes multiplied by the length of the side opposite the angle (which represents the rate at which the ray of light moves along the shore at point x).

Let's denote the rate at which the angle changes as α (alpha). The length of the side opposite the angle can be denoted as y.

Using the concept of similar triangles, we have the following proportion:
(Length of the hypotenuse) / (Length of the side opposite the angle) = (Distance to the lighthouse) / (Distance along the shore)

This can be written as:
300 / y = 300 / x

Now we need to find the relationship between α and x and solve for y.

Since the beacon makes one revolution every 20 seconds, we can say that the angle changes by one complete turn every 20 seconds, or 2π radians every 20 seconds.

Therefore, the rate at which the angle changes, α, is given by:
α = 2π / 20 = π / 10 radians per second

Now, using the proportion 300 / y = 300 / x, we can solve for y:
300 / y = 300 / 200
y = (300 * 200) / 300 = 200 feet

Finally, we can determine the rate at which the ray of light moves along the shore at point x, which is the length of the side opposite the angle:
Rate = α * y = (π / 10) * 200 = 20π / 10 = 2π feet per second

Therefore, the rate at which the ray of light moves along the shore at a point 200 feet from point P is 2π feet per second.

To find the rate at which the ray of light moves along the shore at a point 200 feet from P, we can use the concept of related rates.

Let's denote the angle between the shoreline and the line connecting the lighthouse to point P as θ. Then, the height of the lighthouse above the water surface can be represented by h.

Given that the beacon makes one revolution every 20 seconds, we can determine the relationship between the angle θ and time t as follows:

θ = (2π/20) * t

Now let's assume that x represents the distance along the shoreline from P to the point where we want to find the rate of light movement.

We know that x is 200 feet, and it is changing with time t. Hence, we can write:

x = 200t

To find the rate at which the ray of light moves along the shore, dx/dt, we need to differentiate the expression for x with respect to time t:

dx/dt = d/dt (200t)
= 200

Therefore, the rate at which the ray of light moves along the shore at a point 200 feet from point P is 200 feet per second.