A lighthouse is fixed 130 feet from a straight shoreline. A spotlight revolves at a rate of 11 revolutions per minute, (22 rad/min ), shining a spot along the shoreline as it spins. At what rate is the spot moving when it is along the shoreline 12 feet from the shoreline point closest to the lighthouse?
Concur.
At x=12 ft
dx/dt=(12^2+130^2)/130 * 22π=9062 ft/min
At a given time of t minutes, let the light be x ft down the shore
let the angle formed at the lighthouse with the perpendicular to the shore be Ø
so tanØ = x/130
x = 130tanØ
dx/dt = 130 sec^2 Ø dØ/dt
given dØ/dt = 11(2π) or 22π radian/min (you had 22)
when x = 12
when x = 12, and the other side = 130
hypot^2 = 12^2 + 130^2
hypot = √1704
cosØ = 130/√17044
sec Ø = √17044/130
sec^2 Ø = 17044/16900 = 4261/4225
then dx/dt = 130(4261/4225)(22π)
or appr. .....
you do the button-pushing.
Well, if the spotlight is revolving at a rate of 11 revolutions per minute, it means it completes one revolution in about 5.45 seconds.
Now, to find the rate at which the spot is moving along the shoreline when it is 12 feet away, we can use some light-hearted math. We know that the distance from the lighthouse to the shoreline is fixed at 130 feet, so we can consider this as the radius of our spotlight's circular path.
Let's call the distance from the lighthouse to the spot along the shoreline x. Using the Pythagorean theorem, we find that the distance from the spot to the closest point on the shoreline is sqrt(130^2 - x^2).
Now, let's differentiate both sides of this equation with respect to time (t), which gives us:
0 = 2x * dx/dt - 2t * dt/dt
Since the spotlight is revolving at 22 rad/min, dt/dt is simply 22. Substituting the values and solving for dx/dt, we get:
2x * dx/dt = 2t * 22
Now, when x = 12 feet, we can plug in the values to find dx/dt:
2 * 12 * dx/dt = 2 * 22
24 * dx/dt = 44
dx/dt = 44/24
Simplifying this further, we get:
dx/dt = 11/6 feet per second
So, the spot is moving along the shoreline at a rate of 11/6 feet per second when it is 12 feet away. Keep an eye out for any clownfish swimming by!
To solve this problem, we can use the related rates formula:
rate of change of the spot position along the shoreline = (rate of change of angle) * (distance from the spotlight to the shoreline)
Let's denote:
x = distance along the shoreline from the closest point to the lighthouse
y = distance from the lighthouse to the spotlight
According to the problem, we have:
y = 130 feet (fixed distance)
dx/dt = 12 ft/min (rate at which x is changing)
We want to find dy/dt, the rate at which the spot is moving along the shoreline.
We also know the relationship between x and y:
y^2 = x^2 + 130^2
Differentiating both sides of the equation with respect to time t, we get:
2y(dy/dt) = 2x(dx/dt)
Now we can solve for dy/dt:
dy/dt = (x/y)(dx/dt)
Substituting the given values:
x = 12 ft
y = 130 ft
dx/dt = 12 ft/min
dy/dt = (12/130)(12) = 12/10 = 1.2 ft/min
Therefore, the spot is moving along the shoreline at a rate of 1.2 feet per minute when it is 12 feet from the shoreline point closest to the lighthouse.
To find the rate at which the spot is moving along the shoreline, we can use related rates. Let's assume that the lighthouse is at the origin with coordinates (0, 0) and the shoreline is the x-axis. The spotlight shines a spot along the shoreline as it revolves.
We are given the following information:
- The distance between the lighthouse and the shoreline is 130 feet.
- The spotlight revolves at a rate of 11 revolutions per minute (or 22 radians per minute).
Let's denote the position of the spotlight as (x, 0) on the shoreline, where x is the distance from the lighthouse to the spot.
We want to find the rate at which the spot is moving along the shoreline, dx/dt, when it is 12 feet from the point on the shoreline closest to the lighthouse.
To start, let's find a relationship between x and the angle that the spotlight has swept out. The angle in radians, θ, can be determined by using the fact that the spotlight completes 11 revolutions per minute (or 22 radians per minute). Therefore, in terms of time t, we have:
θ = 22t
Next, let's find a relationship between x and θ. The arc length along the circular path traced by the spotlight is equal to the distance x. The circumference of the circular path is the distance 130 feet. Therefore, we can express the relationship as:
x = θ * (130/2π)
Simplifying, we have:
x = 65θ/π
Now, we can differentiate both sides of the equation with respect to time t:
dx/dt = (d/dt)(65θ/π)
Using the chain rule, we get:
dx/dt = (65/π) * (dθ/dt)
Since we know that dθ/dt is the rate of revolution of the spotlight in radians per minute (22 radians per minute), we substitute this value:
dx/dt = (65/π) * 22
Simplifying further, we get:
dx/dt ≈ 141.692 ft/min
So, the spot is moving along the shoreline at a rate of approximately 141.692 feet per minute when it is 12 feet from the shoreline point closest to the lighthouse.