A security camera needs to be set so that the 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 for the camera. the doorway is 28m from the edge of the parking lot. what angle of view is needed for the camera
My sketch is a triangle with sides 16, 24 and 28 , and I am looking for the angle Ø opposite the 28
by cosine law
28^2 = 16^2+24^2 - 2(16)(24)cosØ
768cosØ = 48
cosØ = 48/76
Ø = 50.8°
Well, if we turn this problem into a game show, we could call it "Angle of View Madness"! Here's the deal, the camera is like a contestant on the show, and it needs to capture the area from the doorway to the edge of the parking lot.
To calculate the angle of view, we need to use some math magic. Let's start by drawing a diagram to visualize the situation. We have a camera in the center, a doorway 16m away, and the edge of the parking lot 24m away. The doorway itself is 28m away from the edge of the parking lot.
Now, imagine the camera is a detective with a magnifying glass, trying to zoom in on the action. The angle of view is essentially how wide the magnifying glass needs to be to see everything from the doorway to the edge of the parking lot.
Using the power of trigonometry, we can determine the angle. We'll use the tangent function: tan(theta) = opposite/adjacent.
In this case, the opposite side is half the distance between the doorway and the edge of the parking lot (28m/2 = 14m), and the adjacent side is the distance from the camera to the doorway (16m).
So, tan(theta) = 14m/16m.
Now, let's break out our calculators and compute this. The angle comes out to be approximately 39.8 degrees.
Therefore, the camera needs an angle of view of about 39.8 degrees to capture the area from the doorway to the edge of the parking lot. Just be sure to let the camera know it's now a game show contestant with a fancy angle as its prize!
To determine the angle of view needed for the camera, we can use trigonometry.
Let's label the points as follows:
- Point A: Camera
- Point B: Doorway
- Point C: Edge of parking lot
From the given information, we know the distances AB = 16m, AC = 24m, and BC = 28m.
We can use the Law of Cosines to find the angle of view. The formula is:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, we want to find angle C, which represents the angle of view. So, we need to rearrange the formula:
cos(C) = (a^2 + b^2 - c^2) / (2ab)
Now, let's substitute the values into the formula:
cos(C) = (16^2 + 28^2 - 24^2) / (2 * 16 * 28)
cos(C) = (256 + 784 - 576) / 896
cos(C) = 464 / 896
cos(C) ≈ 0.51875
Now, to find the angle of view, we can take the inverse cosine (cos^-1) of 0.51875:
C ≈ cos^(-1)(0.51875)
C ≈ 59.8 degrees
Therefore, the angle of view needed for the camera is approximately 59.8 degrees.
To determine the required angle of view for the camera, we can use the concept of trigonometry. Let's break down the problem and go step by step.
1. Draw a diagram: Draw a diagram representing the situation. Label the camera as "C," the doorway as "D," and the edge of the parking lot as "P." Also, label the distances given: Doorway to camera distance (CD) = 16m, Edge of parking lot to camera distance (CP) = 24m, and Doorway to parking lot edge distance (DP) = 28m.
2. 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 lengths of those sides, multiplied by the cosine of the included angle. In this case, we can use it to find the angle at the camera needed to cover the entire area.
The formula for the Law of Cosines is:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, we want to find angle C.
3. Substitute the given values into the formula: Let's assign the following variables:
- Side a = CD = 16m
- Side b = CP = 24m
- Side c = DP = 28m
- Angle C = the angle of view we want to find (in radians)
Now we can substitute these values into the Law of Cosines formula:
DP^2 = CD^2 + CP^2 - 2 * CD * CP * cos(C)
Plugging in the values:
28^2 = 16^2 + 24^2 - 2 * 16 * 24 * cos(C)
4. Solve for the angle: Rearrange the equation to solve for cos(C):
784 = 256 + 576 - 768 * cos(C)
768 * cos(C) = 256 + 576 - 784
768 * cos(C) = 48
cos(C) = 48 / 768
cos(C) = 0.0625
Now, use the inverse cosine function (cos^(-1)) to find the angle in radians:
C = cos^(-1)(0.0625)
Using a calculator or a trigonometric table:
C ≈ 1.508 rad
5. Convert the angle to degrees (optional): If you prefer the angle in degrees, you can convert it by multiplying the radian value by 180/π (approximately 57.3):
C_degrees ≈ 1.508 * (180/π) ≈ 86.5°
Therefore, the camera needs an angle of view of approximately 1.508 radians or 86.5° to include the area from the doorway to the edge of the parking lot.