Answer:
The angle θ, that the device needs to scan is 16.795°
Step-by-step explanation:
Here we have
Height of room = 11 ft
Width of room = 15 ft
Length of room = 18 ft
Position of camera = 6 inches below the ceiling = 11 ft - 6 in = 126 in = 10.5 ft
Distance from camera to edge of long side of the room is given by the following relation;
Long edge of angle = √((18 ft)²+(10.5 ft)²) = 20.839 ft
Shorter edge of angle = √((15 ft)²+(10.5 ft)²) = 18.31 ft
Opposite side of required angle = √((18 ft)²+(15 ft)²) = 23.431 ft
Therefore, by cosine rule, we have
a² = b² + c² - 2·b·c·cos A
We therefore put our a as the opposite side of the required angle, A so we can easily solve for it
our b and c are then the other two sides
23.431² = 20.839² + 18.31² - 2×20.839×18.31×cosA
∴ cos(A) = (23.431² - (20.839² + 18.31²))÷(2×20.839×18.31)
cos(A) = 220.5/763.12418 = 0.29
A = cos⁻¹0.29 = 16.795°
The angle θ, that the device needs to scan = 16.795°.
