Area = area of full circle * (θ/360)
=πr^2*θ/360
=π 50^2*192/360
=4000/3 ft^2.
=πr^2*θ/360
=π 50^2*192/360
=4000/3 ft^2.
Area = area of full circle * (θ/360)
=πr^2*θ/360
=π 50^2*192/360
= 4188.79 ft^2
Volume of cone = (1/3)πr²h
Now, the trick is to determine the radius (r) of the base of the cone. We divide the range of motion (192 degrees) by 360 (a full circle) to get the fraction of the circle covered by the sensor. Then, we multiply that fraction by the circumference of a full circle, 2πr, to find the arc length of the base of the cone.
Arc length = (192/360) * 2πr = (8/15) * 2πr
Once we know the arc length, we can calculate the radius (r) of the base of the cone by using the formula for the circumference of a circle:
Circumference of base = 2πr
(8/15) * 2πr = 50ft
Now we can solve for r. I'll leave that part to you, and once you find the value of r, you can substitute it back into the volume of cone formula to find the area covered by the sensor. Good luck! And don't worry, I'm here for a laugh if it gets too complicated!
1. Field of View (FOV): The sensor has a range of motion of 192 degrees. This means it can detect motion within a 192-degree arc.
2. Detection Distance: The sensor can detect motion up to a distance of 50 feet in front.
To calculate the area, we can visualize it as a sector of a circle.
3. Calculate the Angle of the Sector: To find the angle of the sector covered by the sensor, we need to divide the FOV by 360 degrees (a full circle) and then multiply by 2π (radians).
angle = (FOV / 360) * 2π
= (192 / 360) * 2π
≈ 3.1847 radians
4. Calculate the Radius of the Sector: To find the radius of the sector, we can consider the detection distance as the length of the arc of the sector.
arc length = detection distance = 50 ft
The formula to find the radius (r) of a sector is:
arc length = radius * angle
Substituting the values, we get:
50 = r * 3.1847
r ≈ 15.71 ft
5. Calculate the Area of the Sector: The formula to find the area (A) of a sector is:
A = (1/2) * radius² * angle
Substituting the values, we get:
A = (1/2) * 15.71² * 3.1847
≈ 250.76 ft²
Therefore, the sensor will detect motion and become illuminated over an area of approximately 250.76 square feet.
First, let's understand the dimensions of the coverage area. The sensor light can detect motion for a distance of 50ft in front and has a range of motion of 192 degrees.
To visualize this, imagine a 192-degree angle extending outwards from the sensor light, with a radius of 50ft. The coverage area will be in the shape of a sector of a circle.
To calculate the area of this sector, we can use the formula:
Area of a sector = (θ/360) * π * r^2
Where θ is the angle in degrees, π is a mathematical constant (approximately 3.14159), and r is the radius.
Plugging in the values, we have:
θ = 192 degrees
r = 50ft
Area of the sector = (192/360) * 3.14159 * (50^2)
Simplifying this equation, we get:
Area of the sector = (192/360) * 3.14159 * 2500
Calculating this, the area of the sector becomes approximately 5,236.28 square feet.
So, the sensor light will detect motion and become illuminated over an area of approximately 5,236.28 square feet.