A searchlight rotates at a rate of 4 revolutions per minute.?

The beam hits a wall located 13 miles away and produces a dot of light that moves horizontally along the wall. How fast (in miles per hour) is this dot moving when the angle between the beam and the line through the searchlight perpendicular to the wall is pi/6? Note that dtheta/dt=4(2pi)=8pi.

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To find the speed of the dot of light on the wall, we can use the concept of angular velocity and trigonometry.

Let's start by calculating the angular velocity of the searchlight. The problem states that the searchlight rotates at a rate of 4 revolutions per minute. We know that one revolution is equal to 2π radians, so the angular velocity, ω, can be calculated as follows:

ω = (4 revolutions/minute) * (2π radians/revolution) = 8π radians/minute

Now, let's convert the angular velocity from minutes to hours:

1 hour = 60 minutes

ω = (8π radians/minute) * (60 minutes/hour) = 480π radians/hour

Next, we need to find the rate at which the angle between the beam and the line perpendicular to the wall is changing, dθ/dt. The problem states that dθ/dt = 8π radians/hour.

We can now use trigonometry to relate the angular velocity to the linear speed of the dot of light on the wall. Since the beam hits the wall 13 miles away, we can consider a right triangle formed by the searchlight, the dot of light on the wall, and the line perpendicular to the wall. The hypotenuse of this triangle represents the distance between the searchlight and the dot of light on the wall.

Let's denote the distance between the searchlight and the dot of light on the wall as x miles. Using trigonometry, we can determine that:

tan (π/6) = (x/13)

tan (π/6) = sqrt(3)/3

Now we can find x:

x = 13 * (sqrt(3)/3) = 13sqrt(3)/3 miles

Finally, we can use the relationship between angular velocity and linear speed to find the speed of the dot of light on the wall. The linear speed, v, is equal to the product of the distance from the searchlight to the dot of light on the wall and the angular velocity:

v = x * ω

v = (13sqrt(3)/3 miles) * (480π radians/ hour) = 2080πsqrt(3) miles/hour

Therefore, the speed of the dot of light on the wall is 2080πsqrt(3) miles per hour.