a laser printer is placed on a platform that rotates at a rate of 20rev/min. the beam hits a wall 8m away producing a dot of light that moves horizontally along the wall. let tetabe the angle between the bean and producing the line through the search light perpendicular to the wall. how fast is that dot moving when teta = pie/6 ?
As usual, draw a diagram. If x is the distance of the beam from the perpendicular line to the wall, then
x/8 = tanθ
x = 8tanθ
So,
dx/dt = 8sec^2θ dθ/dt
When θ = π/6 (and that's pi, not pie!!)
dx/dt = 8(4/3)(20*2π) = 1280π/3 m/s
Well, that's quite a mouthful of a question, but don't worry, I'm here to clown around and help you out!
First off, let me convert those units to something a bit more digestible. So, 20 rev/min means that the platform rotates 20 times every minute. Now, let's tackle the problem.
When we look at the situation, we can see that the laser beam is emitting light in the shape of a circle, with the wall being the radius of that circle (which is 8m in this case). The beam rotates around the center of the circle, so the angle "teta" will change as the beam moves.
Now, to figure out how fast that dot of light is moving, we need to use some trigonometry. The speed of the dot moving horizontally along the wall is equal to the rate of change of the angle "teta" multiplied by the radius of the circle.
In this case, we know the rate of change of the angle, which is given as 20 rev/min. But we need to convert that to radians per minute, since we'll be using radians for "teta". Remember that there are 2π radians in one revolution, so we can convert 20 rev/min to (20 * 2π) radians/min.
Since you specifically asked for when "teta" is π/6, we can substitute that value into our equation along with the radius of the circle (8m) to find the speed of the dot at that moment.
So, multiplying the rate of change (20 * 2π) radians/min by the radius (8m) gives us the speed of the dot at that moment.
And on a final note, "pie" is a delicious dessert, but in mathematics, we actually use the symbol "π" (pi).
I hope that wasn't too cheesy of an explanation!
To find the rate at which the dot of light moves along the wall, we'll need to determine the angular velocity of the dot as it rotates with the platform.
Given:
- The platform rotates at a rate of 20 revolutions per minute, which can be converted to angular velocity (ω) in radians per minute.
- The beam hits a wall 8 meters away.
- We need to find the rate of change of the dot's horizontal position, which is represented by the parameter dot dx/dt.
First, let's calculate the angular velocity (ω) in radians per minute:
1 revolution = 2π radians
So, the angular velocity (ω) is:
ω = 20 rev/min * 2π rad/rev
ω = 40π rad/min
Next, we need to find the relationship between θ (theta) and x (the horizontal position of the dot). Based on the given information, we can determine that the angle θ is constant, as the beam hits a fixed spot on the wall.
We can use the trigonometric relationship between the angle θ and the distance x:
x = 8m * tan(θ)
Now, we need to differentiate both sides of the equation with respect to time (t) using the chain rule.
d(x)/dt = d/dt [8m * tan(θ)]
The derivative of 8m is 0, as it is a constant. Therefore, we only need to differentiate tan(θ) with respect to t.
d(x)/dt = 8m * d(tan(θ))/dt
To find d(tan(θ))/dt, we use the derivative of the tangent function:
d(tan(θ))/dt = sec^2(θ) * dθ/dt
Now, we substitute the given value of θ = π/6 into the equation.
sec^2(θ) = sec^2(π/6) = (2/√3)^2 = 4/3
Therefore:
d(x)/dt = 8m * (4/3) * dθ/dt
We are given that d(θ)/dt is the angular velocity (ω). So, we can substitute in the value for ω:
d(x)/dt = 8m * (4/3) * ω
Now, substitute the value of ω:
d(x)/dt = 8m * (4/3) * 40π rad/min
Simplifying:
d(x)/dt = 1280π/3 m/min
Thus, the dot of light is moving at a rate of 1280π/3 meters per minute when θ = π/6.
To find the speed of the dot of light on the wall, we need to consider the rotational speed of the platform and the angle θ (theta) between the beam and the line perpendicular to the wall.
First, let's convert the rotational speed of the platform from revolutions per minute (rev/min) to radians per second (rad/s).
1 revolution = 2π radians
1 minute = 60 seconds
So, the platform's rotational speed in radians per second is:
ω = (20 rev/min) * (2π rad/rev) / (60 sec/min) = 2π/3 rad/s
Next, we'll determine the angular speed of the beam. Since the beam is rotating with the platform, its angular speed is the same as the platform's angular speed, which is ω = 2π/3 rad/s.
Now, let's consider the dot of light on the wall. The dot of light moves horizontally along the wall as the platform rotates. The speed of the dot of light on the wall is given by the perpendicular component of the velocity of the beam.
The perpendicular component of the velocity of the beam can be found using trigonometry. Since the beam is at an angle θ (theta) with the line perpendicular to the wall, the perpendicular component is given by:
v_perpendicular = v * sin(θ)
We need to find the rate at which the dot of light is moving horizontally (v_perpendicular) when θ = π/6.
Let's substitute the values into the equation:
v_perpendicular = v * sin(π/6)
The sine of π/6 is 1/2, so the equation simplifies to:
v_perpendicular = v * 1/2 = v/2
Therefore, when θ = π/6, the dot of light is moving at half the speed of the beam.
Since we already know the angular speed of the beam is ω = 2π/3 rad/s, we can calculate the speed v of the beam using the formula:
v = r * ω
Here, r represents the distance from the laser printer to the wall, which is given as 8m.
Substituting the values into the equation:
v = (8m) * (2π/3 rad/s) = 16π/3 m/s
Finally, substituting the value of v into the equation v_perpendicular = v/2, we can find the speed of the dot of light on the wall when θ = π/6:
v_perpendicular = (16π/3 m/s) / 2 = 8π/3 m/s
Therefore, when θ = π/6, the dot of light is moving at a speed of 8π/3 m/s horizontally along the wall.