Question

Three security cameras were mounted at the corners of a triangular parking lot. camera 1 was 110 ft. from camera 2, which was 137 ft from camera 3. cameras 1 and 3 were 158 ft. apart. which camera had to cover the greatest angle?

Answer 1

Using the Law of Cosines, we can calculate the angles in the triangle determined by the distances between the security cameras. The camera opposite the greatest angle will need to cover the most ground. By performing these calculations, we can identify which camera has to cover the greatest angle.

Step-by-step explanation:

The student is asking about the angles in a triangle, which is determined by the distances between the three security cameras. To determine which camera has to cover the greatest angle, we can use the Law of Cosines, a principle in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles.

Step-by-Step Explanation:

1. Firstly, label the cameras and their corresponding angles. Let's assign Camera 1 to angle A, Camera 2 to angle B, and Camera 3 to angle C.
2. Using the Law of Cosines, we can calculate the angle opposite each side of the triangle. For example, to find angle A, we use the formula: cos(A) = (b2 + c2 - a2) / (2bc), where a, b, and c are the lengths of the sides opposite angles A, B, and C, respectively.
3. Perform this calculation for each angle using the provided distances:
   
   • For angle A (between cameras 2 and 3): a = 158 ft, b = 110 ft, c = 137 ft.
   • For angle B (between cameras 1 and 3): b = 137 ft, a = 110 ft, c = 158 ft.
   • For angle C (between cameras 1 and 2): c = 110 ft, a = 137 ft, b = 158 ft.
4. After calculating each angle, the largest calculated angle will be the angle covering the greatest field of view, meaning that camera will need to cover the greatest angle.

Answer 2

Camera 2 covers the most. If you use your trig rules, you will discover that Camera 2's angle is the largest.