A patrol car is parked 50 feet from a long warehouse The light on the car turns at a rate of 30 revolutions per minute. How fast is the light beam moving along the wall when the beam makes the fol lowing angles with the line perpendicular from the light to the wall?
(a)=30° (b)= 60° (c)=70°
If x is the distance from the closest point on the wall,
x/50 = tanθ
1/50 dx/dt = sec^2(θ) dθ/dt
To find the rate at which the light beam is moving along the wall, we can use trigonometry. Let's first define some variables:
Let:
- x = distance along the wall from the point where the light beam hits the wall.
- θ = angle formed between the line perpendicular from the light to the wall and the line connecting the car to the point where the light beam hits the wall.
We know that the light on the car turns at a rate of 30 revolutions per minute. This means that in one minute, the light completes 30 revolutions or 30 cycles.
To calculate the rate at which the light beam is moving along the wall, we need to find the linear speed of the light beam. The linear speed of an object moving in a circle can be calculated using the formula:
v = rω
Where:
- v is the linear speed of the object,
- r is the radius of the circular path, and
- ω is the angular speed of the object (in radians per unit of time).
In this case, the radius of the circular path is the distance from the car to the point where the light beam hits the wall, which is 50 feet.
To calculate ω, we can convert the number of revolutions per minute to radians per minute since the angular speed is typically measured in radians.
Since one revolution equals 2π radians, we can write:
ω = (30 revolutions/minute) * (2π radians/1 revolution)
Simplifying this expression, we get:
ω = 60π radians/minute
Now, we can calculate the linear speed of the light beam for each angle.
(a) θ = 30°:
For this case, the light beam is forming a right triangle with the wall, where the angle between the wall and the line connecting the point where the light beam hits the wall to the car is 30°.
Using trigonometry, we know that:
sin(θ) = opposite/hypotenuse
In this case, the opposite side is x (the distance along the wall) and the hypotenuse is 50 feet (the radius of the circular path).
sin(30°) = x/50
Rearranging the equation, we get:
x = 50 * sin(30°)
Calculating this value, we find:
x = 25 feet
Now, to find the rate at which the light beam is moving along the wall, we differentiate both sides of the equation:
dx/dt = d(50 * sin(30°))/dt
Since the right side of the equation is a constant (50 * sin(30°) = 25), the derivative of a constant with respect to time is zero. Therefore, the rate at which the light beam is moving along the wall is 0 feet per minute when the angle is 30°.
(b) θ = 60°:
For this case, using the same reasoning, we find that the distance along the wall (x) is:
x = 50 * sin(60°) = 43.3 feet
Differentiating as before, we get:
dx/dt = d(50 * sin(60°))/dt = 0 feet per minute.
(c) θ = 70°:
Following the same steps, we find:
x = 50 * sin(70°) ≈ 45.8 feet
Differentiating, we have:
dx/dt = d(50 * sin(70°))/dt = 0 feet per minute.
Therefore, for all three given angles (30°, 60°, and 70°), the rate at which the light beam is moving along the wall is 0 feet per minute.
To find the speed of the light beam along the wall at different angles, we can use trigonometry. We'll use the fact that the angle of the light beam with the perpendicular line is the same as the angle between the light beam and the wall.
We'll start by finding the length of the light beam as it hits the wall for each angle.
(a) For angle 30°:
The length of the light beam hitting the wall can be found using trigonometry:
sin(30°) = length of light beam / distance from the patrol car to the warehouse
0.5 = length of light beam / 50
length of light beam = 0.5 * 50 = 25 feet
(b) For angle 60°:
Similar to (a), we can use trigonometry:
sin(60°) = length of light beam / 50
√3/2 = length of light beam / 50
length of light beam = (√3/2) * 50 = 25√3 feet
(c) For angle 70°:
Again, we can use trigonometry:
sin(70°) = length of light beam / 50
0.9397 = length of light beam / 50
length of light beam = 0.9397 * 50 = 46.985 feet (approx)
Now, we can find the rate at which the light beam is moving along the wall by differentiating the length of the light beam with respect to time. The light turns at a rate of 30 revolutions per minute, so the angular speed is 30 * 2π radians per minute.
(a) For angle 30°:
The rate at which the light beam is moving along the wall can be found using trigonometry:
rate of change of length = angular speed * length of light beam * cos(30°)
= (30 * 2π) * 25 * cos(30°)
= 150π * √3 feet per minute (approx)
(b) For angle 60°:
Similarly:
rate of change of length = (30 * 2π) * (25√3) * cos(60°)
= 150π * √3 feet per minute (approx)
(c) For angle 70°:
Similarly:
rate of change of length = (30 * 2π) * (46.985) * cos(70°)
= 280.137π feet per minute (approx)
Therefore, the speed of the light beam along the wall is approximately:
(a) 150π * √3 feet per minute,
(b) 150π * √3 feet per minute,
(c) 280.137π feet per minute.