A rotating light is located 25 feet from a wall. The light projected on the wall is moving at a rate of 1.5 feet per second when the light's angle is 15 degrees from perpendicular to the wall. If the light is turning at a constant rate, how many seconds does it take to go around once?
How do I convert to seconds?
To convert from an angle to seconds, we need to understand the relationship between angles and time. In this case, we are given that the rotating light moves at a constant rate, so we can use the formula for angular velocity.
Angular velocity (ω) is the rate at which an object rotates around a particular axis, and it is measured in radians per unit time. To convert from degrees to radians, we use the conversion factor: π/180.
In this problem, we are given the angle in degrees and asked to find the time in seconds. Let's use the formula for angular velocity:
ω = Δθ / Δt
where ω is the angular velocity, Δθ is the change in angle, and Δt is the change in time.
We know that the light's angle changes by 360 degrees (one complete rotation) when it goes around once. So Δθ = 360 degrees.
To find Δt, we need to relate the angle to the distance traveled. Since the light is located 25 feet from the wall, the distance traveled by the light projected on the wall is the circumference of a circle with a radius of 25 feet. The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.
C = 2π(25) = 50π feet
Now we have the relationship between the angle and the distance traveled. The angle is related to the arc length (distance) by the formula:
θ = arc length / radius
In this case, the arc length is the distance traveled by the light projected on the wall, which is 50π feet, and the radius is 25 feet. Therefore, the angle is:
θ = (50π / 25) radians
Now we can substitute these values into the formula for angular velocity:
ω = (360 degrees) / Δt
ω = (50π / 25) radians / Δt
We are given that the rotating light moves at a rate of 1.5 feet per second when the angle is 15 degrees from perpendicular to the wall. Therefore:
1.5 = (50π / 25) / Δt
To solve for Δt, we can rearrange the equation:
Δt = (50π / 25) / 1.5
Now we can calculate this value:
Δt = (50π / 25) / 1.5
Δt = 2π / 1.5
Δt ≈ 4.18879 seconds
Therefore, it takes approximately 4.18879 seconds for the light to go around once.