Question
A source of laser light sends rays AB and AC toward two opposite walls of a hall. The light rays strike the walls at points B and C, as shown below:
A source of laser light is at point A on the ground between two parallel walls. The walls are perpendicular to the ground. AB is a ray of light that strikes the wall on the left at point B. The length of AB is 60m. AC is a ray of light thath strikes the wall on the right at point C, which is 40m above the ground. The ray AB makes an angle of 60 degrees with the ground. The ray AC makes an angle of 45 degrees with the ground.
What is the distance between the walls?
30 m
40 m
70 m
100 m
A source of laser light is at point A on the ground between two parallel walls. The walls are perpendicular to the ground. AB is a ray of light that strikes the wall on the left at point B. The length of AB is 60m. AC is a ray of light thath strikes the wall on the right at point C, which is 40m above the ground. The ray AB makes an angle of 60 degrees with the ground. The ray AC makes an angle of 45 degrees with the ground.
What is the distance between the walls?
30 m
40 m
70 m
100 m
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To find the distance between the two walls, we need to calculate the horizontal distances from point A to points B and C, respectively.
### 1. Calculate Horizontal distance from A to B:
- We know that ray AB makes an angle of \(60^\circ\) with the ground.
- The length of AB is 60m.
Using the cosine function to find the horizontal distance to wall 1 (point B):
\[
d_{AB} = AB \cdot \cos(60^\circ)
\]
\[
d_{AB} = 60 \cdot 0.5 = 30 \text{ m}
\]
### 2. Calculate Horizontal distance from A to C:
- The height of point C is 40m, and ray AC makes an angle of \(45^\circ\) with the ground.
We can use the sine function to find the length of ray AC:
Using the opposite side (height) and the angle \(45^\circ\):
\[
h_C = AC \cdot \sin(45^\circ)
\]
Since we know the height \(h_C = 40m\):
\[
40 = AC \cdot \frac{\sqrt{2}}{2}
\]
Solving for AC:
\[
AC = 40 \cdot \frac{2}{\sqrt{2}} = 40\sqrt{2} \approx 56.57 \text{ m}
\]
Now, using the cosine function to find the horizontal distance to wall 2 (point C):
\[
d_{AC} = AC \cdot \cos(45^\circ)
\]
\[
d_{AC} = 40\sqrt{2} \cdot \frac{\sqrt{2}}{2} = 40 \text{ m}
\]
### 3. Calculate the total distance between the walls:
The distance between the two walls is the total horizontal distance from point A to point C minus the distance from point A to point B:
\[
\text{Distance between the walls} = d_{AB} + d_{AC} = 30 + 40 = 70 \text{ m}
\]
Thus, the distance between the walls is **70 m**.
### 1. Calculate Horizontal distance from A to B:
- We know that ray AB makes an angle of \(60^\circ\) with the ground.
- The length of AB is 60m.
Using the cosine function to find the horizontal distance to wall 1 (point B):
\[
d_{AB} = AB \cdot \cos(60^\circ)
\]
\[
d_{AB} = 60 \cdot 0.5 = 30 \text{ m}
\]
### 2. Calculate Horizontal distance from A to C:
- The height of point C is 40m, and ray AC makes an angle of \(45^\circ\) with the ground.
We can use the sine function to find the length of ray AC:
Using the opposite side (height) and the angle \(45^\circ\):
\[
h_C = AC \cdot \sin(45^\circ)
\]
Since we know the height \(h_C = 40m\):
\[
40 = AC \cdot \frac{\sqrt{2}}{2}
\]
Solving for AC:
\[
AC = 40 \cdot \frac{2}{\sqrt{2}} = 40\sqrt{2} \approx 56.57 \text{ m}
\]
Now, using the cosine function to find the horizontal distance to wall 2 (point C):
\[
d_{AC} = AC \cdot \cos(45^\circ)
\]
\[
d_{AC} = 40\sqrt{2} \cdot \frac{\sqrt{2}}{2} = 40 \text{ m}
\]
### 3. Calculate the total distance between the walls:
The distance between the two walls is the total horizontal distance from point A to point C minus the distance from point A to point B:
\[
\text{Distance between the walls} = d_{AB} + d_{AC} = 30 + 40 = 70 \text{ m}
\]
Thus, the distance between the walls is **70 m**.
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