The problem you're describing involves using trigonometry to determine the height of the cloud cover. We'll model the situation as a right triangle, where the angles and distances give us the information needed to find the height.

Variables:

• The worker shines a light at an angle of $75^\circ$ from the horizontal.
• The observer is $D = 600 \, \text{m}$ away and measures an angle of elevation of $45^\circ$.
• We are looking for the height $h$ of the cloud cover.

Step-by-Step Solution:

1. Form a right triangle:

   • The worker, the observer, and the spot of light form a right triangle, where the distance between the worker and the observer is $D = 600 \, \text{m}$, and the angle of elevation from the observer to the light is $45^\circ$.
   • The height $h$ of the cloud cover forms the vertical leg of the triangle.

2. Use trigonometry:

   • In a right triangle, the tangent of an angle is the ratio of the opposite side (in this case, $h$) to the adjacent side (which is $D$). $\tan(45^\circ) = \frac{h}{600}$ Since $\tan(45^\circ) = 1$: $1 = \frac{h}{600}$ Therefore: $h = 600 \, \text{m}$

Thus, the height $h$ of the cloud cover is approximately 600 meters.

Additional Exploration:

Would you like further clarification, or do you want to see the reasoning behind this result?

Here are 5 related questions that can expand your understanding:

1. How would the height of the cloud cover change if the observer were closer to the worker?
2. What would happen to the height if the angle of elevation measured by the observer was greater than $45^\circ$?
3. How would you calculate the height if the worker's spotlight angle was not $75^\circ$, but some other value?
4. Can you derive a formula for the height $h$ based on varying distance $D$ and angle of elevation?
5. What if the observer measured a different angle for the elevation; how would you adjust the calculation?

Tip: In problems like this, always ensure that the angles and distances are positioned correctly in the triangle to avoid errors in setup.