A lighthouse is located on a small island 4 km away from the nearest

point P on a straight shoreline. Its light makes 3 revolutions
per minute. How fast is the beam of light moving along the shoreline
when it is 1.7 kilometers from
P

?

Draw a diagram. If x is the distance from P, and θ is the angle swept out by the beam, measured from the line joining the lighthouse and P,

x/4 = tan θ

dx/4 = sec^2 θ dθ

dx = 4 sec^2 θ * 3 * 2π

So, what is sec θ? tan θ = 1.7/4 = 0.425
sec^2θ = 1+tan^2θ = 1.18

dx = 4*1.18*3*2π = 88.97 mi/min

To find the speed of the beam of light along the shoreline when it is 1.7 kilometers from point P, we can use related rates.

Let's denote the distance of the beam of light from P as x (in km) and the distance from the lighthouse to P as y (in km).

From the given information, we know that y = 4 km and the light makes 3 revolutions per minute.

We want to find dx/dt, the rate at which x is changing with respect to time.

Since the light is rotating, we can consider the angle theta between the line connecting the lighthouse to P and the line connecting the lighthouse to the beam of light. As the light rotates, theta is changing with time.

Let's denote the rate of change of theta with respect to time as d(theta)/dt (in radians per minute).

Now, we can use trigonometry to relate x, y, and theta.

From the triangle formed by the lines connecting the lighthouse to P, and the lighthouse to the beam of light, we have:

cos(theta) = x / y

Differentiating both sides of the equation with respect to time t, we get:

-d(theta)/dt * sin(theta) = dx/dt / y

Since we know that d(theta)/dt is constant (since the light makes 3 revolutions per minute), we can write it as w, the angular velocity in radians per minute.

So, the equation becomes:

-w * sin(theta) = dx/dt / y

Now, we need to find theta when x = 1.7 km.

From the triangle formed by the lines connecting the lighthouse to P, and the lighthouse to the beam of light, we have:

sin(theta) = x / (y + 1.7)

Substituting this value into the previous equation, we get:

-w * (x / (y + 1.7)) = dx/dt / y

Rearranging the equation, we have:

dx/dt = -w * x * y / (y + 1.7)

Now, we can substitute the known values:

dx/dt = -3 * (1.7) * 4 / (4 + 1.7)

Simplifying the expression, we get:

dx/dt = -20.4 / 5.7

Therefore, the speed of the beam of light along the shoreline when it is 1.7 kilometers from point P is approximately -3.579 km/min.

To find the speed of the beam of light along the shoreline when it is 1.7 kilometers from point P, we can use the concepts of relative motion and trigonometry.

Let's first define some variables:
- Let x represent the distance between the beam of light and point P along the shoreline.
- Let y represent the distance between the lighthouse and point P along the shoreline.
- Let r represent the distance between the lighthouse and point P (which is 4 km in this case).
- Let θ represent the angle between the line connecting the lighthouse, point P, and the line perpendicular to the shoreline.

We can create a right triangle with these variables, where x, y, and r form the sides of the triangle. The angle θ is opposite to the side x.

Using trigonometry, we have:
sin(θ) = x / r

Differentiating both sides of this equation with respect to time, we get:
cos(θ) * dθ/dt = dx/dt / r

We also know that the lighthouse makes 3 revolutions per minute. Since each revolution corresponds to a full circle (2π radians), the angular speed of the lighthouse is:
dθ/dt = (3 revolutions / 1 minute) * (2π radians / 1 revolution) * (1 minute / 60 seconds)

Now we can substitute the given values:
r = 4 km = 4000 m
x = 1.7 km = 1700 m

To find θ, we can use the Pythagorean theorem:
r^2 = x^2 + y^2
4000^2 = 1700^2 + y^2
y = sqrt(4000^2 - 1700^2)

Now we can substitute all these values into the equation involving the differentiation:
cos(θ) * [(3 * 2π) / (60 seconds)] = dx/dt / 4000

Finally, we solve for dx/dt:
dx/dt = [cos(θ) * (3 * 2π) / (60 seconds)] * 4000

Evaluate cos(θ) using the value of y we found earlier, and substitute it into the equation to get the desired answer.