Three security cameras were mounted at the corners of a triangular parking lot. Camera 1 was 122 ft from camera 2, which was 139 ft from camera 3. Cameras 1 and 3 were 151 ft apart. Which camera had to cover the greatest angle?(1 point)
Responses
Camera 1
Camera 2
Camera 3
There is not enough information to tell.
To determine which camera has to cover the greatest angle, we can analyze the triangle formed by the distances between the cameras. We have a triangle where:
- The distance between Cameras 1 and 2 is 122 ft.
- The distance between Cameras 2 and 3 is 139 ft.
- The distance between Cameras 1 and 3 is 151 ft.
According to the Law of Cosines, c^2 = a^2 + b^2 - 2ab * cos(C), where a and b are the sides of the triangle and C is the angle opposite the side c.
We need to calculate the angle at each camera (each vertex of the triangle) using the distances provided.
For the angle at Camera 1 (let's call it angle A), the opposing side is the one between Cameras 2 and 3, which is 139 ft.
For the angle at Camera 2 (let's call it angle B), the opposing side is the one between Cameras 1 and 3, which is 151 ft.
For the angle at Camera 3 (let's call it angle C), the opposing side is the one between Cameras 1 and 2, which is 122 ft.
Using the Law of Cosines to calculate each angle:
For angle A:
cos(A) = (122^2 + 151^2 - 139^2) / (2 * 122 * 151)
cos(A) = (14884 + 22801 - 19321) / (36742)
cos(A) = (18364) / (36742)
cos(A) ≈ 0.499593108
A ≈ cos^(-1) (0.499593108)
A ≈ 60.1 degrees
For angle B:
cos(B) = (122^2 + 139^2 - 151^2) / (2 * 122 * 139)
cos(B) = (14884 + 19321 - 22801) / (33908)
cos(B) = (11404) / (33908)
cos(B) ≈ 0.336297604
B ≈ cos^(-1) (0.336297604)
B ≈ 69.9 degrees
For angle C:
cos(C) = (139^2 + 151^2 - 122^2) / (2 * 139 * 151)
cos(C) = (19321 + 22801 - 14884) / (41958)
cos(C) = (27238) / (41958)
cos(C) ≈ 0.649330906
C ≈ cos^(-1) (0.649330906)
C ≈ 49.9 degrees
From the calculated angles, we can see that the greatest angle is at Camera 2 (angle B), which is approximately 69.9 degrees.
Therefore, Camera 2 has to cover the greatest angle.