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
To determine which camera had to cover the greatest angle, we need to find the angles formed by the three cameras.
Let's label the angles as follows:
- Angle A: Angle formed by cameras 1 and 2.
- Angle B: Angle formed by cameras 2 and 3.
- Angle C: Angle formed by cameras 3 and 1.
To find the angles, we can use the Law of Cosines. According to the Law of Cosines, the squared length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of those sides multiplied by the cosine of the included angle.
Using this law, we can find the angles:
For Angle A:
122^2 = 139^2 + 151^2 - 2(139)(151) * cos(A)
14884 = 19321 + 22801 - 41878*cos(A)
-47618*cos(A) = -27038
cos(A) = -27038 / -47618
cos(A) ≈ 0.5673
A ≈ acos(0.5673)
A ≈ 55.86°
For Angle B:
139^2 = 122^2 + 151^2 - 2(122)(151) * cos(B)
19321 = 14884 + 22801 - 41878*cos(B)
-47618*cos(B) = -13396
cos(B) = -13396 / -47618
cos(B) ≈ 0.2812
B ≈ acos(0.2812)
B ≈ 73.87°
For Angle C:
151^2 = 139^2 + 122^2 - 2(139)(122) * cos(C)
22801 = 19321 + 14884 - 33898*cos(C)
-33898*cos(C) = -1346
cos(C) = -1346 / -33898
cos(C) ≈ 0.0397
C ≈ acos(0.0397)
C ≈ 88.93°
Now we have the measures of the three angles:
Angle A ≈ 55.86°
Angle B ≈ 73.87°
Angle C ≈ 88.93°
Among these angles, Angle C (formed by cameras 3 and 1) is the largest with a measure of approximately 88.93°. Therefore, camera 3 had to cover the greatest angle.