A lighthouse is located on a small island 4 km away from the nearest point P on a straight shoreline and its light makes two revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? (Round your answer to one decimal place.)
To solve this problem, we can use related rates, which involves finding the rate of change of one quantity with respect to another.
Let's denote the distance along the shoreline as x and the distance from the lighthouse to the beam of light as y.
We're given that the lighthouse is located 4 km away from the nearest point P on the shoreline, so we have a right triangle with the lighthouse at one vertex, the light beam at another vertex, and the point P at the third vertex.
Using the Pythagorean theorem, we can express the relationship between x and y as follows:
x^2 + y^2 = 4^2
Differentiating both sides of this equation with respect to time (t), we obtain:
2x(dx/dt) + 2y(dy/dt) = 0
Since we're interested in finding how fast the beam of light is moving along the shoreline when it is 1 km from P, we need to determine dx/dt when x = 1 km.
We also know that the light makes two revolutions per minute, which means it completes 2 * 360 degrees per minute. So, the angular speed (ω) of the light is:
ω = (2 rev/min) * (360 degrees/rev) * (π/180 degrees)
Now, we can relate the angular speed to the rate of change of y (dy/dt) and solve for dy/dt:
dy/dt = ω * y
Substituting the known values, we get:
dy/dt = [(2 rev/min) * (360 degrees/rev) * (π/180 degrees)] * y
To find dy/dt when y = 1 km, we can substitute y = 1 km into the above equation and solve for dy/dt:
dy/dt = [(2 rev/min) * (360 degrees/rev) * (π/180 degrees)] * (1 km)
Calculating this expression will give us the rate at which the beam of light is moving along the shoreline when it is 1 km from point P.
To find the speed of the beam of light along the shoreline when it is 1 km from point P, we can use the concept of related rates.
Let's assume that the lighthouse is located at point A on the island, and point P is on the shoreline. We want to find the rate at which the beam of light is moving along the shoreline when it is 1 km from P.
First, let's set up some variables:
- Let x represent the distance between the beam of light and point P along the shoreline.
- Let y represent the distance between the lighthouse and point P (4 km in this case).
- Let θ represent the angle between the line connecting the lighthouse and point P, and the line connecting the lighthouse and the beam of light.
We need to find dx/dt, which represents the rate at which x is changing with respect to time.
From the given information, we can see that the angle θ is changing at a constant rate of 2 revolutions per minute. Since there are 60 seconds in a minute, this means that dθ/dt = 2 * 360 degrees / 60 seconds = 12 degrees per second.
To relate the variables x, y, and θ, we can use trigonometry. From the right triangle formed by the lines connecting the lighthouse, point P, and the beam of light, we can write the equation:
tan(θ) = x / y
We can differentiate this equation implicitly with respect to time (t) to find the relationship between dx/dt, dy/dt, and dθ/dt:
sec^2(θ) * dθ/dt = (dy/dt * x - y * dx/dt) / y^2
Since y is a constant (4 km), dy/dt = 0. Thus, the equation simplifies to:
sec^2(θ) * dθ/dt = -dx/dt / y
To find dx/dt, we need to find the value of sec^2(θ) at the given position.
Let's examine the right triangle formed by the lines connecting the lighthouse, point P, and the beam of light when the beam is at a distance of 1 km from P. In this case, the triangle is a right triangle with sides x and y = 4 km.
Using the Pythagorean theorem, we can find x:
x^2 + y^2 = 1^2
x^2 + 4^2 = 1
x^2 = 1 - 16
x^2 = -15 (No solution for x; this is not possible)
Since x has no solution, this means that when the beam of light is 1 km from point P, it is not along the shoreline.
Therefore, there is no answer to this question.