A lighthouse 2 miles offshore has a light that rotates once every 20 seconds. At what rate is the light traveling along the shore if it is shining on a point 4 miles along the shore from the point nearest the lighthouse?
Draw a diagram. It is clear that when the distance of the beam from the point P on shore nearest the lighthouse is x,
tanθ = x/2
sec^2θ dθ/dt = 1/2 dx/dt
when x=4, we have
(1+(4/2)^2)(2π/20) = 1/2 dx/dt
dx/dt = π mi/s
To find the rate at which the light is traveling along the shore, we need to use trigonometry and differentiate the given information.
Let's denote the angle between the line connecting the lighthouse to the point on the shore and the line connecting the lighthouse to the point nearest it on the shore as θ. We can determine θ using trigonometry.
In this case, we have a right triangle where the hypotenuse is the line connecting the lighthouse to the point on shore (4 miles), the adjacent side is the line connecting the lighthouse to the point nearest it on the shore (2 miles), and the opposite side is the distance the light travels in one rotation (the circumference of a circle with radius 2 miles or 4π miles).
Using the cosine function, we have:
cos(θ) = adjacent / hypotenuse
cos(θ) = 2 / 4
cos(θ) = 1/2
Now, we need to find the relationship between the angle θ and the time it takes for the light to rotate once. The time it takes for the light to complete one rotation is given as 20 seconds.
Since the light rotates 360 degrees in 20 seconds, the angular velocity (ω) is:
ω = 2π radians / 20 seconds
ω = π / 10 radians/second
Now, we can determine the rate at which the light travels along the shore by differentiating the position function with respect to time.
Let's denote the rate at which the light travels along the shore as v (velocity). We have:
v = dθ/dt * ds/dθ
where s represents the distance along the shore.
To find dθ/dt (the rate at which the angle changes with respect to time), we can take the derivative of cos(θ) with respect to time:
d(cos(θ))/dt = 0
Since the cosine function is a constant (1/2), its derivative is zero.
Therefore, dθ/dt = 0.
Next, we need to find ds/dθ (the rate at which the distance along the shore changes with respect to θ).
We can use trigonometry to relate ds/dθ to the side lengths of the triangle. We have:
ds/dθ = adjacent / cos(θ)
ds/dθ = 2 / (1/2)
ds/dθ = 4
Now, we can calculate v:
v = dθ/dt * ds/dθ
v = 0 * 4
v = 0
So, the rate at which the light is traveling along the shore is 0.