A lighthouse is located on a small island 2.5 km from the nearest point on a straight shoreline. Its light makes 3 revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 kilometers from the point directly opposite the lighthouse
draw a picture
let Θ be the angle between the line to the nearest point and the light beam
... s is the distance from the nearest point to the light beam
tan(Θ) = s / 2.5 km
dΘ/dt = 6 π / min
differentiating ... d[tan(Θ)]/dt = .4 ds/dt ... sec^2(Θ) dΘ/dt = .4 ds/dt
ds/dt = 2.5 sec^2(Θ) dΘ/dt = 2.5 km * [(2.5^2 + 1^2) / 2.5^2] * 6 π / min
To solve this problem, we can use the concept of related rates. We need to find the rate at which the beam of light is moving along the shoreline when it is 1 kilometer (1000 meters) from the point directly opposite the lighthouse.
Let's denote the distance along the shoreline as x (in meters), and the distance from the lighthouse to that point as y (in meters). We are given that y = 2.5 km (or 2500 meters). We need to find dx/dt, the rate at which x is changing with respect to time.
To solve related rates problems, we typically need to establish a relationship between the variables involved. In this case, we can use the Pythagorean theorem:
x^2 + y^2 = d^2
where d is the distance from the lighthouse to the point on the shoreline where the beam hits. We are given that d = 1 km (or 1000 meters). Now, let's differentiate this equation with respect to time:
2x(dx/dt) + 2y(dy/dt) = 2d(dd/dt)
Since we are interested in dx/dt, we can rearrange the equation as follows:
2x(dx/dt) = -2y(dy/dt) + 2d(dd/dt)
We know the values of y, d, and dy/dt. y = 2500 meters, d = 1000 meters, and dy/dt is unknown but given that the light makes 3 revolutions per minute.
Now, let's substitute these values into the equation and solve for dx/dt:
2x(dx/dt) = -2(2500)(3 revolutions per minute) + 2(1000)(0) [Note: dd/dt is zero since the distance from the lighthouse to the point on the shoreline is constant]
Simplifying the equation, we get:
2x(dx/dt) = -15000
Now we can solve for dx/dt:
dx/dt = -15000 / (2x)
Since we want to find the value of dx/dt when x = 1000 meters, we can substitute this into the equation:
dx/dt = -15000 / (2 * 1000)
= -7.5 meters per minute
Thus, the beam of light is moving at a rate of -7.5 meters per minute along the shoreline when it is 1 kilometer from the point directly opposite the lighthouse. The negative sign indicates that the beam of light is moving in the opposite direction along the shoreline.