A lighthouse stands 450 m off a straight shore and the focused beam of its light revolves four times each minute. As shown in the figure, P is the point on shore closest to the lighthouse and Q is a point on the shore 250 m from P. What is the speed of the beam along the shore when it strikes the point Q? Describe how the speed of the beam along the shore varies with the distance between P and Q. Neglect the height of the lighthouse.
Let x be the distance PQ. We want to find how fast Q is moving -- that is, dx/dt
If you draw the diagram, you can see that when the angle θ = PLQ,
tanθ = x/450
so
sec^2θ dθ/dt = 1/450 dx/dt
and since the light rotates 4 times per minute, one sweep takes 15 seconds. So dθ/dt = 2π/15 rad/s
Now plug all that in to find dx/dt when x=250
tanθ = 250/450 = 5/9
so sec^2θ = 1 + 25/81 = 106/81
106/81 * 2π/15 = 1/450 dx/dt
x = 2120π/27 ≈ 246.67 m/s
Oh, we're talking about lighthouses now? I guess you could say they really "shine" in their line of work. Alright, let's shed some light on this question.
So, we know that the lighthouse stands 450 m off the shore and the beam revolves four times in a minute. That's quite the spin cycle! Now, the distance between P and Q is 250 m.
To find the speed of the beam along the shore when it strikes point Q, we can use a bit of trigonometry. The beam's speed is basically its angular speed (how fast it revolves) multiplied by the distance between the lighthouse and point Q.
The angular speed can be calculated by taking the number of revolutions per minute and converting it to radians per second. In this case, the beam makes 4 revolutions per minute, so that's 4 times 2π radians divided by 60 seconds.
Now, we multiply the angular speed by the distance between the lighthouse and Q. That's 4 times 2π divided by 60, multiplied by 250.
Calculating it all, we get the speed of the beam along the shore when it strikes point Q. But let me tell you, the speed is going to vary based on the distance between P and Q. The farther away Q is from P, the faster the beam is going to speed along the shore. It's like the beam is saying, "I wanna go fast, baby!" It's all about that distance, my friend.
So, there you have it! The speed of the beam along the shore when it strikes point Q depends on the distance between P and Q. I hope I didn't leave you in the dark with that answer!
To find the speed of the beam along the shore when it strikes point Q, we can consider the circular path that the beam follows.
Let's assume that the lighthouse is at the center of the circle and the beam is the radius. Since the light revolves four times each minute, it completes four full circles in one minute.
The circumference of a circle can be calculated using the formula C = 2πr, where C is the circumference and r is the radius. In this case, the circumference of the circular path that the beam follows is 4 times the circumference of the circle with a radius of 450 m.
C = 4 * 2π * 450
C = 3600π meters
Now, we need to find the time it takes for the beam to travel from point P to point Q. Since the beam completes four full circles in one minute, it means that it takes 1/4th of a minute (or 15 seconds) for it to reach point Q from point P.
To find the speed, we divide the distance traveled by the time taken:
Speed = Distance / Time
Distance = length of arc from P to Q
The length of an arc can be calculated using the formula L = θr, where L is the length of the arc, θ is the angle in radians, and r is the radius.
In this case, the angle θ is 90 degrees (since the beam travels from P to Q, which is a quarter of a circle) and converted to radians, it becomes π/2.
L = (π/2) * 450
L = (π * 450)/2
L = 225π meters
So, the speed of the beam along the shore when it strikes point Q is given by:
Speed = (225π meters) / (15 seconds)
Speed ≈ 15π meters/second
Hence, the speed of the beam along the shore when it strikes point Q is approximately 15π meters/second.
The speed of the beam along the shore varies linearly with the distance between points P and Q. As we can see, the length of the arc from P to Q is directly proportional to the distance between P and Q. Therefore, if the distance between P and Q is doubled, the length of the arc and the speed of the beam will also double.
To find the speed of the beam along the shore when it strikes point Q, we need to consider the distance the beam travels along the shore in one revolution and the number of revolutions per minute.
Let's assume that one full revolution of the beam covers a distance of D meters along the shore.
Given that the beam revolves four times each minute, the distance covered by the beam along the shore in one minute would be 4D.
Now, let's consider the figure provided.
P is the point on the shore closest to the lighthouse, and Q is a point on the shore 250 m from P. This means the distance between P and Q is 250 m.
Since one full revolution covers a distance of D along the shore, the beam at P completes one-fourth of its revolution. Therefore, the distance the beam travels along the shore from P to Q is (1/4)D.
The distance (1/4)D is equal to 250 m. Hence, we can write the equation:
(1/4)D = 250
To find D, we can multiply both sides of the equation by 4:
D = 250 * 4
D = 1000 m
So, one full revolution of the beam covers a distance of 1000 meters along the shore.
Now, to find the speed of the beam along the shore when it strikes point Q, we need to determine the time taken for the beam to travel from P to Q.
Since the beam revolves four times per minute, it takes 1/4th of a minute (or 15 seconds) for the beam to complete one revolution. Therefore, the time taken for the beam to travel from P to Q is 15 seconds.
Using the formula speed = distance / time, we can find the speed of the beam along the shore when it strikes point Q:
Speed = (1/4)D / 15
Speed = (1/4) * 1000 / 15
Speed = 250 / 15
Speed ≈ 16.67 m/s
So, the speed of the beam along the shore when it strikes point Q is approximately 16.67 meters per second.
As for how the speed of the beam along the shore varies with the distance between P and Q, we can observe that the speed is constant. No matter how close or far Q is from P, the beam always completes one revolution in the same amount of time (15 seconds), resulting in a constant speed of approximately 16.67 m/s along the shore.