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.
velocity= dtheta/dt * distance
in the units you have, with distance in miles, and dt in minutes, velocity will be in miles/minute
θπ
cos(π/6)=sqrt(3)/2
sec^2(π/6)=4/3
dθ/dt=4 rev/min = 8π rad/min
tan(θ)=x/13
x=13 tan(θ)
dx/dt=13 sec^2(θ) dθ/dt
At θ=π/6
dx/dt=13*(4/3)*8π=435.6 miles/min = 26138 miles/hr
Note to above attempt: velocity in x-direction, not radial velocity.
Well, well, well, looks like we have a revolving searchlight and a moving dot of light. Quite the shining situation, isn't it?
Now, let's get to solving this puzzle. We have the angular speed of the searchlight, which is 8π radians per minute. But we need to find the speed of the dot in miles per hour when the angle between the beam and the perpendicular line is π/6.
First, we need to find the linear speed of the dot on the wall. To do that, we need to use some trigonometry. The distance between the searchlight and the wall is 13 miles, so the length of the arc on the wall covered by the dot will be 13π/6 miles (since the angle is π/6).
Now, to find the linear speed of the dot, we multiply the length of the arc by the angular speed. So, the linear speed is (13π/6) * (8π) = 104π²/3 miles per minute.
But, hold your horses, we need the speed in miles per hour. So, let's convert that. There are 60 minutes in an hour, so the speed is (104π²/3) * 60 = 6240π²/3 miles per hour.
And there you have it, when the angle between the beam and the perpendicular line is π/6, the dot on the wall is moving at a speed of approximately 6240π²/3 miles per hour.
Remember, while the math may be serious, the journey of light can always brighten our day with a little humor!
To find the speed of the dot of light moving along the wall, we can use the concept of related rates.
Given:
- The rate of rotation of the searchlight is 4 revolutions per minute, which is equal to 8π radians per minute (since 1 revolution is equal to 2π radians).
- The angle between the beam and the line through the searchlight perpendicular to the wall is π/6.
Let's break down the problem into smaller steps:
Step 1: Determine the angular velocity (dθ/dt) of the searchlight in radians per minute.
- In the problem, it states that the searchlight rotates at a rate of 4 revolutions per minute, which we already converted to 8π radians per minute.
Step 2: Determine the linear velocity (dx/dt) of the dot of light moving along the wall.
- As the searchlight rotates, the dot of light on the wall moves horizontally.
- We want to find the speed of the dot of light, which is dx/dt.
Step 3: Relate the angular velocity and the linear velocity.
- The angle between the beam and the line through the searchlight perpendicular to the wall is π/6.
- The tangent of this angle is equal to the ratio of dx/dt (linear velocity) to dθ/dt (angular velocity).
- So we have the equation: tan(π/6) = dx/dt / dθ/dt.
Now, let's solve for dx/dt:
1. Given : dθ/dt = 8π radians per minute
2. We can rewrite the equation as: tan(π/6) = dx/dt / (8π)
3. Solve for dx/dt:
- Multiply both sides of the equation by (8π):
(8π) * tan(π/6) = dx/dt
- Simplify:
dx/dt = 8π * tan(π/6)
- Calculate the value:
dx/dt ≈ 18.396 miles per minute
To convert the speed from minutes to hours:
- Multiply the speed by 60 minutes per hour:
dx/dt ≈ 18.396 * 60 ≈ 1103.76 miles per hour
Therefore, the dot of light is moving at approximately 1103.76 miles per hour along the wall.