A light in a lighthouse 5 kilometers offshore from a straight shoreline is rotating at 4 revolutions per minute. How fast is the beam moving along the shoreline when it passes the point 5 kilometers from the point opposite the​ lighthouse?

Make a sketch.

Mine has a straight shore line and a lighhouse L and a point A on the shore so that LA is perpendicular to the shore.
I let P be a point on the shore so that AP = x
Let Ø be the angle between LA and LP

given: dØ/dt = 4(2π) radians/min
= 8π rads/min

tanØ = x/5
x = 5tanØ
dx/dt = 5sec^2 Ø dØ/dt
when Ø = 0
dx/dt = 5 (sec^2 0)(8π) km/min
= 5(1)(8π) km/min
=40π km/min

change it to whatever units are needed.

To solve this problem, we can use trigonometry and the concept of angular velocity.

Let's denote:
- d as the horizontal distance from the lighthouse to the point on the shoreline where the beam passes. This distance will decrease as the beam rotates.
- r as the distance from the lighthouse to the point at which the beam passes 5 kilometers offshore from the opposite point on the shoreline.

We can observe that as the beam rotates, it creates a right triangle, where r is the hypotenuse, and d is the base of the triangle.

To find the speed of the beam moving along the shoreline, we need to find the rate of change of d with respect to time.

First, we need to convert the given angular velocity from revolutions per minute to radians per second since we'll be using radians in trigonometry. There are 2π radians in one revolution, and 60 seconds in one minute, so the angular velocity can be calculated as follows:

Angular velocity = (4 revolutions per minute) * (2π radians per revolution) / (60 seconds per minute) = (4 * 2π) / 60 radians per second = (8π/60) radians per second = (2π/15) radians per second.

Now, let's differentiate the equation of the triangle to find the relationship between d and r. Since r is a constant value (5 kilometers), we can write the equation as:

r^2 = d^2 + 5^2.

To find the rate of change of d with respect to time, we differentiate this equation implicitly:

2r * (dr/dt) = 2d * (dd/dt).

Simplifying this equation, we get:

r * (dr/dt) = d * (dd/dt).

Now, let's substitute the known values:

5 * (dr/dt) = d * (dd/dt).

To determine the value of d, we can use the trigonometric relationship between the angle θ and d:

sinθ = d / r.

Rearranging this equation, we get:

d = r * sinθ.

Since r is constant and equal to 5 kilometers, we can differentiate both sides of this equation with respect to time:

dd/dt = r * d( sinθ )/dt.

We know that d( sinθ )/dt is the rate at which the beam rotates, which is the angular velocity, so we can substitute:

dd/dt = 5 * (2π/15) = (10π/15) kilometers per second.

Now, we can substitute this value back into the equation we derived earlier:

5 * (dr/dt) = d * (dd/dt).

Plugging in the values:

5 * (dr/dt) = 5 * (10π/15).

Simplifying this equation, we get:

(dr/dt) = (2π/3) kilometers per second.

Therefore, the beam is moving along the shoreline at a speed of (2π/3) kilometers per second when it passes the point 5 kilometers from the point opposite the lighthouse.

To determine the speed at which the light beam is moving along the shoreline, we need to use some trigonometry. Let's break down the problem step by step:

1. Determine the circumference of the path traced by the light beam:
Since the light is rotating at 4 revolutions per minute, we know that it completes a full circle every 1/4 minute (4 revolutions = 1 circle). The circumference of this circle is the distance traveled by the light beam in one revolution.

To calculate the circumference, we can use the formula:
Circumference = 2 * π * radius

In this case, the radius of the circle is the distance from the lighthouse to a point on the shoreline, which is 5 kilometers.

Circumference = 2 * π * 5 = 10π kilometers

2. Determine the distance the light beam travels along the shoreline in one minute:
To find the distance the light beam travels along the shoreline in one minute, we need to calculate the arc length of the path traced by the beam during that time.

Arc Length = (angle in degrees / 360) * Circumference

Since the light beam completes one revolution in 1/4 minute, the angle swept by the beam is 360 degrees.

Arc Length = (360/360) * (10π) = 10π kilometers

3. Determine how fast the light beam is moving along the shoreline:
Now that we know the distance the light beam travels along the shoreline in one minute (10π kilometers), we can use this value to determine the speed.

Speed = Distance / Time

Since the time is one minute, the speed is equal to the distance traveled along the shoreline, which we calculated as 10π kilometers.

Hence, the speed of the light beam along the shoreline when it passes the point 5 kilometers from the point opposite the lighthouse is 10π kilometers per minute.