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The problem asks for the speed at which a fire is spreading in the direction of a ranger who is watching from a mesa 3,201 feet high. The ranger observes the change in the angle of depression of the leading edge of the fire from 11.31° to 13.56° over 5 minutes. We are asked to calculate the speed of the fire in miles per hour.

### Approach:
We can use trigonometry, particularly the tangent function. The height of the mesa (3,201 feet) acts as the opposite side in a right triangle, and the distance of the fire from the base of the mesa is the adjacent side.

1. **Set up the tangent equation** for both angles:
   - For the initial angle (11.31°): 
     \[
     \text{tan}(11.31^\circ) = \frac{3201}{d_1}
     \]
     where \(d_1\) is the initial distance from the fire.
   - For the final angle (13.56°):
     \[
     \text{tan}(13.56^\circ) = \frac{3201}{d_2}
     \]
     where \(d_2\) is the final distance.

2. **Find the change in distance**:
   \[
   \Delta d = d_1 - d_2
   \]
   This is the distance the fire has traveled in 5 minutes.

3. **Convert the speed to miles per hour**:
   First, find the speed in feet per minute:
   \[
   \text{Speed in feet/min} = \frac{\Delta d}{5}
   \]
   Then, convert to miles per hour by using:
   \[
   \text{Speed in mph} = \text{Speed in feet/min} \times \frac{60}{5280}
   \]

Let’s calculate this. I'll go through each step.The speed at which the fire is spreading in the direction of the ranger is **6.2 miles per hour**.

Would you like further details on any part of the solution or help with related concepts?

Here are 5 related questions you might find useful:
1. How does trigonometry help in calculating real-world distances like in this problem?
2. What are other practical uses of the tangent function?
3. How would the calculation change if the ranger was on a different mesa height?
4. What if the fire's angle of depression changed over a different time period?
5. How can we estimate the fire's speed if the angles were measured in smaller increments?

**Tip:** When solving problems involving angles of elevation or depression, always visualize the right triangle formed by the observer's line of sight, the height of the object, and the ground distance to simplify calculations.

A forest ranger atop a 3201-ft mesa is watching the progress of a forest fire spreading in her direction. In 5 minutes the angle of depression of the leading edge of the fire changed from 11.31° to 13.56°. At what speed in miles per hour is the fire spreading in the direction of the ranger?

This problem involves using the tangent function in trigonometry to calculate the speed of a spreading forest fire. A ranger observes the change in the angle of depression over 5 minutes, allowing us to compute the fire's speed in miles per hour.

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