A lighthouse beacon rotates through 360 degrees every 12 seconds, it is located 100 m off the shore of an island with a coastline of steep cliffs running north and south. As the light beam sweeps clockwise, starting from the north (direction of 0 degrees), it strikes some part of the cliff.
How long does it take the light beam to reach 30 degrees? What is the distance travelled by the beam to the cliff at that time?
I know how to find the amount of time it takes which is 1 second, I I found out the period which is
p=360/12s
=30
and then I took the reciprocal of that 1/30 and multiplied it by 30 degrees and got 1 second.
I just don't know how to find the distance it travels at 30 degrees at 1 second. I suppose it has to do with the Y. I've been stuck on this for days now!
THANKS for clarity.
To find the distance traveled by the light beam when it reaches 30 degrees, we need to divide the circumference of the circle (which represents 360 degrees) by 360 and then multiply it by 30. This will give us the fraction of the total distance traveled by the beam.
The circumference of a circle is given by the formula: C = 2πr, where r is the radius of the circle. In this case, the radius of the circle represents the distance from the lighthouse to the cliff, which is 100 meters.
Using the formula, we can calculate the circumference:
C = 2πr = 2π(100) = 200π meters
To find the distance traveled at 30 degrees, we can use the fraction of the total distance:
Distance traveled at 30 degrees = (30/360) * 200π = (1/12) * 200π
≈ 16.67π meters
So, when the light beam reaches 30 degrees, it will have traveled approximately 16.67π meters.
To find the distance traveled by the light beam when it reaches 30 degrees, we can apply some basic trigonometry.
First, let's consider a right triangle formed by the lighthouse, the point where the light beam hits the cliff, and the point directly below the lighthouse on the coastline. The 30-degree angle we are interested in is one of the acute angles in this triangle.
Since we know the lighthouse is 100 meters off the shore, the side opposite the 30-degree angle (let's call it side Y) is actually the distance traveled by the light beam.
To find this distance, we can use the trigonometric function tangent (tan). The tangent of an angle is defined as the ratio of the opposite side to the adjacent side.
In this case, we can use the tangent formula:
tan(30 degrees) = Y / 100 meters
Now, we can solve for Y:
Y = tan(30 degrees) * 100 meters
To calculate this, you can use a scientific calculator or an online trigonometry calculator to find the tangent of 30 degrees (which is approximately 0.57735). Multiplying it by 100 meters gives us:
Y ≈ 0.57735 * 100 meters
Y ≈ 57.735 meters
Therefore, the distance traveled by the light beam when it reaches 30 degrees is approximately 57.735 meters.
If the distance is r, then
r/30 = cos 30°