A searchlight initially points due west. The light rotates clockwise at a constant speed so that it completes one revolution every 2 minutes. A building is located 200 yards west and 50 yards north of the searchlight. How many minutes into the first revolution of the searchlight will the building be illuminated by the searchlight?
I understand how to get the angle (arctan(3/20)) but I don't know how to get the time. Please help!
if it rotates once in 2 minutes, the angular velocity is
2π/2 = π radians/minute
Not sure how you got tanθ = 3/20
I get tanθ = 50/200 = 1/4
so it will take
arctan(1/4)/π minutes to illuminate the building
Ah, I got the numbers mixed up. Thank you so much!
To determine the time it takes for the searchlight to reach the building, we need to calculate the angle it needs to rotate from the initial west position.
The angle can be found using trigonometry. The tangent of the angle is given by the ratio of the north displacement (50 yards) to the west displacement (200 yards):
tan(angle) = 50/200
angle = arctan(50/200)
angle ≈ 0.24498 radians
Since the searchlight completes one revolution every 2 minutes, the rate of change of the angle can be calculated by dividing the angle of a full revolution (2π radians) by the time it takes (2 minutes):
rate of change of angle = (2π radians) / (2 minutes)
rate of change of angle = π radians/minute
Therefore, to find the time it takes for the searchlight to reach the building, we can divide the angle needed to reach the building (0.24498 radians) by the rate of change of the angle (π radians/minute):
time = angle / rate of change of angle
time = 0.24498 radians / (π radians/minute)
time ≈ 0.078 minutes
So, it will take approximately 0.078 minutes, or about 4.7 seconds, for the searchlight to illuminate the building during its first revolution.
To determine the time it takes for the searchlight to illuminate the building, we need to consider the angular speed and the angle the searchlight needs to rotate to illuminate the building.
In one full revolution of the searchlight, it covers an angle of 360 degrees or 2π radians. Since the searchlight completes one revolution every 2 minutes, we can determine its angular speed as follows:
Angular speed = (2π radians) / (2 minutes) = π/minute
Now, let's determine the angle the searchlight needs to rotate to illuminate the building. We can use the given information that the building is located 200 yards west and 50 yards north of the searchlight.
Using the tangent function, we can find the angle between the searchlight pointing west and the line connecting the searchlight to the building:
tan(θ) = (50 yards) / (200 yards) = 1/4
θ = arctan(1/4)
Now we have the angle (θ) that the searchlight needs to rotate to illuminate the building.
To find the time it takes for the searchlight to rotate to that angle, we can use the formula:
Time = Angle / Angular Speed
Time = (arctan(1/4)) / (π/minute)
Now we can evaluate this expression:
Time = (arctan(1/4)) * (1 / (π/minute)) = (arctan(1/4)) * (minute/π)
We can use a calculator to find the value of arctan(1/4) ≈ 14.036 degrees. Therefore:
Time = (14.036 degrees) * (minute/π)
Now, to convert from degrees to minutes, we can multiply by the conversion factor:
Time = (14.036 degrees) * (minute/π) * (π/180 degrees) = (14.036/180) minutes ≈ 0.078 minutes
Hence, the building will be illuminated by the searchlight approximately 0.078 minutes into its first revolution.