A rotating beacon is located 1 kilometer off a straight shoreline. If the beacon rotates at a rate of 3 revolutions per minute, how fast (in kilometers per hour) does the beam of light appear to be moving to a viewer who is 1/2 kilometer down the shoreline.
I need to show work, so formatting answers in this manner would be most appreciated. Thanks in advance! :) :)
Evaluate the limit as h -> 0 of:
[tan (pi/6 + h) - tan(pi/6)]/h
I thought the answer was √3/3, or tan(pi/6, but apparently that is wrong, any tips here?
Sorry, I mean to make a new question, disregard above 'answer'.
Gee, now you're going to make people who open this topic think the answer has already been provided. Thanks a ton...
You're a jerk.
tan(θ)=x/1
cos(θ)=1/sqrt(x^2+1)
sec^2(θ)=x^2+1
dθ/dt=3 rev/min=6π rad/min
x=tan(θ)
dx/dt=sec^2(θ) dθ/dt
At x=0.5 km
dx/dt=(0.5^2+1)*6π=23.6 km/min=1414 kph
To find the speed at which the beam of light appears to be moving to the viewer, we can use the concept of angular speed and the equation for linear speed.
First, let's convert the rotation rate of the beacon from revolutions per minute to radians per second. We know that 1 revolution is equal to 2π radians. Therefore, the angular speed is (3 rev/minute) * (2π rad/rev) * (1/60 min/sec).
= (3/60) * 2π rad/sec
= π/10 rad/sec.
Next, we can use the formula for linear speed, which relates linear speed, angular speed, and radius. The formula is:
Linear speed = angular speed * radius.
In this case, the radius is 1 kilometer. We need to convert it to meters to remain consistent with the units of the linear speed. Since 1 kilometer is equal to 1000 meters, the radius is 1000 meters.
Finally, we can calculate the linear speed in kilometers per hour. We know that 1 hour is equal to 3600 seconds, and we can use the conversion factor of 1000 meters = 1 kilometer.
Linear speed = (π/10 rad/sec) * (1000 meters) * (3600 sec/hour) / (1000 meters/km)
= (π/10) * (3600) km/hour
≈ (π/10) * 3600 km/hour
≈ 1130.97 km/hour (rounded to two decimal places).
Therefore, the beam of light appears to be moving at a speed of approximately 1130.97 kilometers per hour to the viewer who is 1/2 kilometer down the shoreline.