A sensor light can detect motion for a distance of 70ft. with a range of motion of 189 degrees. Over what area will the sensor detect motion and become illuminated?
a = 1/2 r^2 θ
convert θ to radians and plug in the numbers.
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To determine the area in which the sensor will detect motion and become illuminated, we need to calculate the coverage area of the sensor.
To begin, let's visualize the information we have:
- The sensor can detect motion for a distance of 70ft.
- The range of motion is 189 degrees.
To calculate the coverage area, we can imagine a cone or a fan shape with the sensor at the center. The base of the cone corresponds to the distance the sensor can detect motion, which is 70ft. The angle between the two sides of the cone is given by the range of motion, which is 189 degrees.
To find the area covered by the sensor, we need to calculate the surface area of this cone. However, since the sensor itself is a point, we can consider only the curved surface area.
First, let's convert the range of motion from degrees to radians:
1 degree = π/180 radians
189 degrees = (189 * π/180) radians = (21 * π/20) radians
Next, let's calculate the radius of the base of the cone:
The circumference of the base of the cone is the distance the sensor can detect motion, which is 70ft.
Circumference = 2πr
70ft = 2πr
r = 70ft / (2π)
Now, we can calculate the curved surface area of the cone:
Curved Surface Area = 2πr * l, where l is the slant height of the cone.
To find the slant height, we can use the sine function:
sin(θ) = Opposite / Hypotenuse
sin((21 * π/20)/2) = (70ft/2) / l
l = (70ft/2) / sin((21 * π/20)/2)
Finally, we can calculate the curved surface area:
Curved Surface Area = 2πr * l
Curved Surface Area = 2π * (70ft/(2π)) * [(70ft/2) / sin((21 * π/20)/2)]
Simplifying this expression will give us the area covered by the sensor.