A lighthouse is located on a small island 4 km away from the nearest point P on a straight shoreline and its light makes three 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.)
as usual, draw a diagram and you will see that if θ=0 when x=0,
tanθ = x/4
so,
sec^2θ dθ/dt = 1/4 dx/dt
Now, you have θ and dθ/dt, so just solve for dx/dt
To solve this problem, we can use the chain rule of differentiation.
Let's suppose that the lighthouse is located at point A on the island, and point P is on the shoreline.
We are given the following information:
- The distance from the lighthouse to the nearest point P on the shoreline is 4 km.
- The lighthouse's light makes three revolutions per minute.
To find how fast the beam of light is moving along the shoreline when it is 1 km from point P, we need to find the rate of change of the distance between the beam of light and point P with respect to time.
Let's denote:
- The distance between point P and the beam of light as x (in km).
- The distance between the lighthouse and the beam of light as y (in km).
Using the Pythagorean theorem, we have the equation:
x^2 + y^2 = (4 km)^2
Differentiating both sides of the equation implicitly with respect to time (t), we get:
2x * dx/dt + 2y * dy/dt = 0
Rearranging the equation, we have:
dy/dt = -(x/y) * dx/dt
We are required to find dy/dt when x = 1 km. So, substituting the given values into the equation:
x = 1 km
y = sqrt((4 km)^2 - (1 km)^2) = sqrt(16 km^2 - 1 km^2) = sqrt(15 km^2) = 3.87 km (approx.)
Now, we need to find dx/dt, which represents the rate at which the distance between the beam of light and point P is changing.
The beam of light moves in a circle with a circumference of 2πr, where r represents the distance between the lighthouse and the beam of light (in km). Since the light makes three revolutions per minute, the light completes 3 * 2πr distance per minute. In terms of km per minute, this can be written as (3 * 2πr) km/min.
We know that dy/dt = -(x/y) * dx/dt
So, dx/dt = -(y/x) * dy/dt
Substituting the values:
y = 3.87 km
x = 1 km
dx/dt = -(3.87 km / 1 km) * (3 * 2π * 3.87 km) km/min
dx/dt = -11.6π km^2/min
Now, let's calculate the numeric value of dx/dt:
dx/dt ≈ -11.6 * 3.14 km^2/min
dx/dt ≈ -36.41 km^2/min
Therefore, the beam of light is moving along the shoreline at a speed of approximately 36.4 km^2/min when it is 1 km from point P.
To solve this problem, we can use the concept of related rates.
Let's denote the distance between the lighthouse and point P on the shoreline as x (which changes over time). We are given that the lighthouse makes three revolutions per minute.
We want to find the rate of change of x (or dx/dt) when x is 1 km (or 1,000 m).
To begin, let's establish a relationship between the different variables involved.
Since the lighthouse is 4 km away from the nearest point P on the shoreline, we can form a right triangle using the lighthouse, point P, and the current position of the beam of light (which we'll call point Q).
Using the Pythagorean theorem, we can write:
x^2 + 4^2 = Q^2
Differentiating with respect to time (t), we get:
2x(dx/dt) = 2Q(dQ/dt)
Since we are interested in finding dx/dt when x = 1,000 m, we can substitute x = 1,000 into the equation:
2(1,000)(dx/dt) = 2Q(dQ/dt)
Simplifying, we have:
1,000(dx/dt) = Q(dQ/dt)
Now, we need to find Q and dQ/dt.
We are given that the lighthouse makes three revolutions per minute. In each revolution, the beam of light travels the circumference of a circle, which is equal to 2π times the radius.
The radius can be found as the hypotenuse of the right triangle formed by the lighthouse, point P, and point Q (when x = 1,000), using the Pythagorean theorem:
x^2 + 4^2 = Q^2
(1,000)^2 + 4^2 = Q^2
1,000,000 + 16 = Q^2
Q^2 = 1,000,016
Q ≈ 1,000.008 m (rounded to six decimal places)
Now, we can find dQ/dt by considering that three revolutions occur in one minute:
dQ/dt = (3 revolutions / 1 minute) * (2π * Q)
dQ/dt = 3 * 2 * π * 1,000.008 (substituting the value of Q)
dQ/dt ≈ 18,849 m/min (rounded to the nearest whole number)
Substituting these values into our equation:
1,000(dx/dt) = Q(dQ/dt)
1,000(dx/dt) = 1,000.008 * 18,849
dx/dt ≈ 18.8 m/min (rounded to one decimal place)
Therefore, when the beam of light is 1 km from point P, it is moving along the shoreline at a rate of approximately 18.8 m/min.
At x=1 km
dθ/dt=3 rev/min=6π rad/min
dx/dt=(1^2+4^2)/4 * 6π=80.1 km/min=4806.6 kph