In the problem provided, Ben measures the width of a river using a theodolite from the roof of a 13-meter tall building. The two angles of depression given are 30° and 36°. We can calculate the width of the river using trigonometry by applying the tangent function to these angles.

Steps to solve:

1. Let the width of the river be denoted by $w$, and the distances from the building to the two points on the riverbank as $d_1$ and $d_2$.

2. Use the tangent function with the height of the building (13 m) and the angles to determine the distances:

   • For the angle of 30°, we use: $\tan(30^\circ) = \frac{13}{d_1}$ Solving for $d_1$, we get: $d_1 = \frac{13}{\tan(30^\circ)}$

   • For the angle of 36°, we use: $\tan(36^\circ) = \frac{13}{d_2}$ Solving for $d_2$, we get: $d_2 = \frac{13}{\tan(36^\circ)}$

3. The width of the river $w$ is the difference between $d_1$ and $d_2$, i.e., $w = d_1 - d_2$.

Let me calculate these values.The width of the river is approximately 5 meters.

Would you like further details or explanations on any part of this problem?

Here are five related questions to explore:

1. How would the width change if Ben measured from a higher or lower building?
2. What if the angles of depression were closer together?
3. How could you solve this problem if only one angle were given?
4. How does changing the height of the building affect the measured distances?
5. How could this problem be solved using another trigonometric method?

Tip: Always remember to convert angles to radians when using trigonometric functions in most calculators or coding environments.