A security camera needs to be set so that its angle of view includes the area from a doorway to the edge of a parking lot. The doorway is 16m from the camera. The edge of the parking lot is 24m from the camera. The doorway is 28m from the edge of the parking lot. What angle of view is needed for the camera?
To find the angle of view needed for the camera, we can use the Law of Cosines. The Law of Cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the magnitudes of those two sides multiplied by the cosine of the angle between them.
In this case, we have a triangle with sides of 16m, 24m, and 28m. Let's call the angle between the sides of length 16m and 24m as angle A.
Using the Law of Cosines, we can write the equation as:
28^2 = 16^2 + 24^2 - 2(16)(24) * cos(A)
Simplifying this equation, we get:
784 = 256 + 576 - 768 * cos(A)
Rearranging the equation, we have:
768 * cos(A) = 256 + 576 - 784
768 * cos(A) = 48
cos(A) = 48 / 768
cos(A) = 0.0625
Now, to find the angle A, we can take the inverse cosine (cos^-1) of 0.0625. Using a calculator, we find:
A ≈ cos^-1(0.0625) ≈ 85.01 degrees
Therefore, the angle of view needed for the camera is approximately 85.01 degrees.