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.
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.
1 answer