At night, a security camera pans over a parking lot. The camera is on a post at point a which is 53m from point c and 71 m from point b.the distance from B to c is 68m. Calculate the angle through which the camera pans
54 degrees
To calculate the angle through which the camera pans, we can use trigonometry and the Law of Cosines.
First, let's draw a diagram to illustrate the given information:
```
A
|
| 53m
|
_______C_______
| |
| 68m |
| |
B
```
Now, let's label the angle through which the camera pans as θ.
Using the Law of Cosines, we have the equation:
b^2 = a^2 + c^2 - 2ac * cos(θ)
In this case,
a = 53m
b = 71m
c = 68m
We can substitute these values into the equation:
71^2 = 53^2 + 68^2 - 2 * 53 * 68 * cos(θ)
Now, we can solve for cos(θ). Rearranging the equation, we get:
cos(θ) = (53^2 + 68^2 - 71^2) / (2 * 53 * 68)
By substituting the values and calculating, we find:
cos(θ) = 0.633922261
Finally, to find the angle θ, we can use the inverse cosine function (cos^-1) by taking the arccosine of the calculated value:
θ = cos^-1(0.633922261)
Calculating this, we find:
θ ≈ 50.19 degrees
Therefore, the angle through which the camera pans is approximately 50.19 degrees.