To find the distance between the two walls, we can analyze the positions of points B and C based on the information given about the angles and lengths of the light rays.
First Ray (AB):
- Length of ray AB = 40 m
- Angle with the ground = 60 degrees
By using trigonometric functions, we can determine the vertical and horizontal distances traveled by ray AB:
- Vertical distance to point B (height) can be found using: \[ \text{Height} = AB \times \sin(60^\circ) = 40 \times \frac{\sqrt{3}}{2} \approx 40 \times 0.866 \approx 34.64 , m \]
- Horizontal distance to point B can be found using: \[ \text{Horizontal distance} = AB \times \cos(60^\circ) = 40 \times \frac{1}{2} = 20 , m \]
Second Ray (AC):
- Length of ray AC creates a height of 80 m above the ground.
- Angle with the ground = 45 degrees
Using the height to find the horizontal distance:
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Since this ray achieves a height of 80 m, we can find the length of ray AC (denote this length as x).
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With the 45-degree angle: \[ \sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2} \] Thus, using the vertical height, \[ 80 = x \cdot \sin(45^\circ) \Rightarrow x = \frac{80}{\sin(45^\circ)} = \frac{80}{\frac{\sqrt{2}}{2}} = 80 \cdot \frac{2}{\sqrt{2}} = 80 \cdot \sqrt{2} \approx 113.14 , m \]
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The horizontal distance for ray AC would also be equal to this used with the cosine: \[ \text{Horizontal distance} = AC \cdot \cos(45^\circ) = (80\sqrt{2}) \cdot \frac{\sqrt{2}}{2} = 80 \]
Distance Between the Walls:
- The total distance between the walls is the sum of the horizontal distances from point A to points B and C.
Thus, the distance between the walls is: \[ \text{Distance} = \text{Horizontal distance to B} + \text{Horizontal distance to C} = 20 , m + 80 , m = 100 , m \]
The answer is 100 m.
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