A)A lighthouse emits a beam of light that rotates (counterclockwise from a birds-eye perspective) at a constant rate with one full rotation every 10 seconds. 100 feet east of the lighthouse is a long wall running north-south. How fast is the light’s reflection traveling along the wall northwards when it crosses the point on the wall closest to the lighthouse?
B) A mechanic alters the lighthouse to make the light rotate in such a way that its reflection travels northward along the wall at a constant rate of 100 feet per second. How fast, in radians per second, is the beam of light rotating when the light’s reflection is 100 feet north of the point on the wall that is closest to the lighthouse?
draw a diagram. Label P the point on the wall closest to the lighthouse. If the beam strikes the wall at a distance x from P, such that the angle of the beam is θ, then
x/100 = tanθ
1/100 dx/dt = sec^2θ dθ/dt
we know that dθ/dt = π/5
Now just plug in your numbers to find dx/dt or dθ/dt as needed.
A) To find the speed at which the light's reflection is traveling along the wall northwards when it crosses the point on the wall closest to the lighthouse, we can use the concept of related rates.
Let's assume the distance between the lighthouse and the point on the wall closest to it is x.
Since the lighthouse emits a beam of light that rotates at a constant rate with one full rotation every 10 seconds, the angular velocity (ω) of the light's rotation is 2π/10 radians per second.
The speed at which the light's reflection is traveling along the wall northwards is given by dx/dt, where dx represents the change in x and dt represents the change in time.
Now, let's determine a relationship between x, the distance from the lighthouse to the point on the wall, and θ, the angle the light's rotation has made.
The circumference of a circle with radius x is given by 2πx. Since the light's rotation covers a full circle in 10 seconds, we can write:
2πx = 10
Differentiating both sides of the equation with respect to time (t), we get:
2π(dx/dt) = 0
Simplifying the equation, we find:
dx/dt = 0
Therefore, when the light's reflection crosses the point on the wall closest to the lighthouse, the speed at which it is moving in the northward direction is 0 feet per second.
B) If the mechanic alters the lighthouse so that the light's reflection travels northward along the wall at a constant rate of 100 feet per second, we need to find the rate at which the beam of light is rotating, in radians per second.
Let's assume the distance between the lighthouse and the point on the wall closest to it is y.
The speed at which the light's reflection is traveling northwards along the wall is given by dy/dt = 100 feet per second.
Since the reflection of the light is traveling northward, the angle θ the light's rotation has made is y/x, where x is the same distance as defined in Part A.
To find the rate at which the beam of light is rotating, we need to find dθ/dt (the change in θ with respect to time).
Using trigonometry, we can relate θ with x and y through the tangent function:
tan(θ) = y/x
Differentiating both sides of the equation with respect to time (t), we get:
sec^2(θ) * dθ/dt = (dy/dt * x - y * dx/dt) / x^2
Since we know dx/dt from Part A to be 0, the equation simplifies further:
sec^2(θ) * dθ/dt = (dy/dt * x) / x^2
Substituting the known values, we have:
sec^2(θ) * dθ/dt = (100 * x) / x^2
Simplifying the equation, we find:
dθ/dt = 100 / x
To find x, we can use the relationship obtained in Part A:
2πx = 10
Solving for x, we get:
x = 10 / (2π)
Substituting this value back into the equation for dθ/dt, we have:
dθ/dt = 100 / (10 / (2π))
Simplifying the equation, we find:
dθ/dt = 20π radians per second
Therefore, when the light's reflection is 100 feet north of the point on the wall closest to the lighthouse, the beam of light is rotating at a rate of 20π radians per second.
A) To find the speed of the light's reflection along the wall northwards, we can use the concept of related rates. Let's assume that at a certain moment, the light beam is at a position on its rotational path represented by an angle θ measured in radians.
The lighthouse emits one full rotation every 10 seconds, which corresponds to a rate of 2π radians per 10 seconds or (2π/10) radians per second. This is the rate at which the angle θ is changing.
Now, we need to find the rate at which the reflection of the light is moving northwards along the wall. Let's call this rate dx/dt, where x represents the distance between the lighthouse and the point on the wall closest to it.
Since the lighthouse is located 100 feet east of the wall, the distance x can be considered as the hypotenuse of a right-angled triangle formed by the lighthouse, the point on the wall, and the vertical line from the point on the wall to the lighthouse.
Using trigonometry, we can express x in terms of the angle θ as x = 100 sec(θ) (since sec(θ) = hypotenuse/adjacent = x/100).
Now, differentiating both sides of the equation with respect to time t, we get dx/dt = 100sec(θ) * d(θ)/dt.
To find dx/dt when θ is changing at a rate of (2π/10) radians per second, we substitute these values into the equation:
dx/dt = 100sec(θ) * d(θ)/dt = 100sec(θ) * (2π/10) = 20πsec(θ) feet per second.
Therefore, the speed of the light's reflection along the wall northwards when it crosses the point closest to the lighthouse is 20πsec(θ) feet per second.
B) In this case, the light's reflection is traveling northwards along the wall at a constant rate of 100 feet per second. Let's call this rate dx/dt again.
Since the distance x is changing at a constant rate of 100 feet per second, we have dx/dt = 100.
We want to find the rate at which the beam of light is rotating when the light's reflection is 100 feet north of the point on the wall closest to the lighthouse. Let's call this rate dθ/dt, where θ represents the angle at that particular moment.
Similarly to part A, we can express x in terms of θ as x = 100 sec(θ).
Differentiating both sides with respect to time t, we get dx/dt = 100sec(θ) * d(θ)/dt.
Since dx/dt = 100, we have 100 = 100sec(θ) * d(θ)/dt.
Dividing both sides by 100sec(θ), we get d(θ)/dt = 1/sec(θ).
Since sec(θ) = x/100 = 100/100 = 1, we have d(θ)/dt = 1.
Therefore, the beam of light is rotating at a rate of 1 radian per second when the light's reflection is 100 feet north of the point on the wall closest to the lighthouse.