A searchlight rotates at a rate of 4 revolutions per minute. The beam hits a wall located 9 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 5? Note that ddt=4(2)=8

Question

A searchlight rotates at a rate of 4 revolutions per minute. The beam hits a wall located 9 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 5? Note that ddt=4(2)=8

Answer 1

let x be the angle

let d be the distance of the dot from the point on the wall closest to the light

tan(x) = d/9
sec^2(x) dx/dt = 1/9 dd/dt
dd/dt = 1/9 sec^2(x) dx/dt

dx/dt = 4*2pi/min
I assume x = 5°, since 5 radians would point away from the wall.

dd/dt = 9 * 1.00765 * 25.13274 = 227.843 mi/min * 60min/hr = 13670.6 mi/hr

Check for sanity. If the wall were a circle 9 miles in radius, the circumference of the wall would be 9*2pi = 56.5 miles. That distance would be covered 4 times per minute, making the dot travel at a constant speed of 13572 mi/hr.

That would be the slowest speed observed when traveling along a straight wall, at the instant when the dot is closes to the light. Since 5° is a small angle, we'd expect the speed to be close to that figure, and it is, but slightly faster.

Answer 2

To find the speed at which the dot of light is moving along the wall, we need to relate the angular velocity of the searchlight to the linear velocity of the dot.

First, let's convert the given angular velocity from revolutions per minute to radians per minute. Since one revolution is equal to 2π radians, the angular velocity is 4 revolutions/min * 2π radians/revolution = 8π radians/min.

Now we need to relate the angular velocity to the linear velocity. The linear velocity of the dot along the wall can be represented by the formula:

v = r * ω,

where v is the linear velocity, r is the distance from the searchlight to the wall, and ω is the angular velocity.

In this case, the distance from the searchlight to the wall is given as 9 miles, so r = 9 miles. Therefore, the linear velocity of the dot on the wall is:

v = 9 miles * (8π radians/min).

Now, to convert the velocity from miles per minute to miles per hour, we need to multiply by the conversion factor of 60 minutes per hour:

v = 9 miles * (8π radians/min) * (60 min/hour).

Simplifying the expression:

v = 9 * 8π * 60 miles * radians / (min * hour).

Since we want the answer in miles per hour, we can further simplify:

v ≈ 9 * 8π * 60 ≈ 1,356π miles/hour.

So, the dot of light is moving approximately at a speed of 1,356π miles per hour when the angle between the beam and the line perpendicular to the wall is 5.