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The problem in the image describes a forest fire spreading, where a forest ranger stationed on a 3240-ft mesa observes the angle of depression to the fire change from 11.34° to 13.5° over a 5-minute period. We are asked to find the speed at which the fire is spreading, in miles per hour.

### Step-by-step solution:

1. **Draw a right triangle** to represent the situation, where:
   - The height of the mesa is 3240 feet.
   - The horizontal distance from the fire to the ranger decreases as the fire spreads. The angle of depression changes as well.
   - We need to find how far the fire has moved in 5 minutes.

2. **Using trigonometry** (tangent of the angle of depression), we know:
   \[
   \tan(\theta) = \frac{\text{height of mesa}}{\text{horizontal distance to the fire}}
   \]
   For the two angles, we can set up the following relationships:
   - At \( \theta_1 = 11.34^\circ \): 
     \[
     d_1 = \frac{3240}{\tan(11.34^\circ)}
     \]
   - At \( \theta_2 = 13.5^\circ \):
     \[
     d_2 = \frac{3240}{\tan(13.5^\circ)}
     \]

3. **Calculate the difference in distance** between \(d_1\) and \(d_2\), representing the distance the fire has spread over 5 minutes:
   \[
   \Delta d = d_1 - d_2
   \]

4. **Convert feet to miles** (since 1 mile = 5280 feet):
   \[
   \Delta d_{\text{miles}} = \frac{\Delta d}{5280}
   \]

5. **Find the speed in miles per hour** by converting the time from minutes to hours (5 minutes = \( \frac{5}{60} \) hours):
   \[
   \text{Speed (mph)} = \frac{\Delta d_{\text{miles}}}{\frac{5}{60}} = \Delta d_{\text{miles}} \times \frac{60}{5}
   \]

Let me calculate this now.The fire is spreading at approximately **6.0 miles per hour** in the direction of the ranger.

Do you have any questions or would you like further details on any part of this solution?

Here are 5 related questions to explore:
1. How do we apply the tangent function to solve problems involving right triangles?
2. How does the angle of depression relate to distances in trigonometry?
3. What are other methods for calculating the speed of an object over time?
4. How would the problem change if the height of the mesa was different?
5. How can this method be applied to real-world problems, such as monitoring weather or natural disasters?

**Tip:** In trigonometry problems involving angles of depression or elevation, always sketch a diagram first. This helps visualize the right triangle and apply the correct trigonometric ratios.

A forest ranger atop a 3,240-ft mesa is watching the progress of a forest fire spreading in his direction. In 5 minutes, the angle of depression of the leading edge of the fire changed from 11.34° to 13.5°. At what speed in miles per hour is the fire spreading in the direction of the ranger? Round to the nearest tenth.

Learn how to calculate the speed of a forest fire spreading in the direction of a ranger using trigonometric concepts and angles of depression. The ranger observes a 3240-ft mesa and notes an angle change from 11.34° to 13.5°. Suitable for Grades 9-12.

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