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?

Use Law of Cosines:

a = 24 m.
b = 28 m.
c = 16 m.

CosB = (a^2 + c^2 - b^2)/2ac
CosB = (576+256-784) / 768 = 0.0625
B = 86.4 Deg. = Angle of view.

To determine the angle of view needed for the camera, we can use the concept of trigonometry.

Let's denote:
- The distance from the camera to the doorway as A (16m),
- The distance from the camera to the edge of the parking lot as B (24m),
- The distance from the doorway to the edge of the parking lot as C (28m).

To find the angle of view, we can use the Law of Cosines, which states:

c^2 = a^2 + b^2 - 2ab * cos(C),

where:
- c is the side opposite angle C,
- a and b are the other two sides.

In our case, angle C is the angle of view. So, a = A, b = B, and c = C.

Substituting the given values into the equation, we get:

C^2 = 16^2 + 24^2 - 2 * 16 * 24 * cos(C).

To find the angle C, we need to calculate it using this equation. We can then take the inverse cosine (cos^-1) to get the angle in degrees.

To find the angle of view needed for the camera, we can use trigonometry. We'll use the tangent function because we have the opposite side and the adjacent side.

First, let's label the distances:
- Distance from the camera to the doorway (opposite side) = 16m
- Distance from the camera to the edge of the parking lot (adjacent side) = 24m
- Distance from the doorway to the edge of the parking lot (hypotenuse) = 28m

Now we can use the tangent function:

tan(angle) = (opposite side) / (adjacent side)

Plugging in the values we know:

tan(angle) = 16m / 24m

Now we can solve for the angle:

angle = arctan(16m / 24m)

Using a calculator, we find:

angle ≈ 33.69 degrees

So, the camera needs to have an angle of view of approximately 33.69 degrees to include the area from the doorway to the edge of the parking lot.