Three security cameras were mounted at the corners off a triangular parking lot. Camera 1 was 122 ft from camera 2, which was 139 ft from camera 3. Camera 1 and 3 were 151 ft apart. Which camera had to cover the greatest angle.
To determine which camera had to cover the greatest angle, we need to find the angles between the cameras.
Using the Law of Cosines, we can find the angles:
For Camera 1:
Let the angles be denoted as follows:
A = angle at Camera 1
B = angle at Camera 2
C = angle at Camera 3
We have the following information:
Side a = 122 ft (opposite A)
Side b = 139 ft (opposite B)
Side c = 151 ft (opposite C)
Using the Law of Cosines:
c^2 = a^2 + b^2 - 2ab * cos(C)
151^2 = 122^2 + 139^2 - 2(122)(139) * cos(C)
2(122)(139) * cos(C) = 122^2 + 139^2 - 151^2
cos(C) = (122^2 + 139^2 - 151^2) / (2(122)(139))
cos(C) ≈ 0.468
Using the inverse cosine function (cos⁻¹) to find the angle:
C ≈ cos⁻¹(0.468)
C ≈ 63.1°
For Camera 2:
Using the Law of Cosines:
b^2 = a^2 + c^2 - 2(ac * cos(B))
139^2 = 122^2 + 151^2 - 2(122)(151) * cos(B)
cos(B) = (122^2 + 151^2 - 139^2) / (2(122)(151))
cos(B) ≈ 0.816
Using the inverse cosine function (cos⁻¹) to find the angle:
B ≈ cos⁻¹(0.816)
B ≈ 35.9°
Now that we have found the angles, we can determine which camera had to cover the greatest angle.
The greatest angle is C, which is found at Camera 3. Therefore, Camera 3 had to cover the greatest angle.