1. **Problem Statement:** A fire ranger is at the top of a 90-ft observation tower. He sees two fires: one due west at an angle of depression of 5°, and another due south at an angle of depression of 2°. We want to find how far apart the two fires are. 2. **Understanding Angle of Depression:** The angle of depression is the angle between the horizontal line from the observer's eye and the line of sight down to the object. It is measured downward from the horizontal. 3. **Step 1: Define variables and draw right triangles.** - Height of tower, $h = 90$ ft. - Angle of depression to west fire, $\theta_w = 5^\circ$. - Angle of depression to south fire, $\theta_s = 2^\circ$. 4. **Step 2: Calculate horizontal distances to each fire.** Using right triangle trigonometry, the horizontal distance $d$ to each fire is related to the height and angle of depression by: $$d = h \times \cot(\theta)$$ 5. **Calculate distance to west fire:** $$d_w = 90 \times \cot(5^\circ)$$ 6. **Calculate distance to south fire:** $$d_s = 90 \times \cot(2^\circ)$$ 7. **Step 3: Calculate the distance between the two fires.** Since the fires are due west and due south, the distance between them forms the hypotenuse of a right triangle with legs $d_w$ and $d_s$: $$D = \sqrt{d_w^2 + d_s^2}$$ 8. **Step 4: Compute values:** - $\cot(5^\circ) = \frac{1}{\tan(5^\circ)} \approx \frac{1}{0.08749} \approx 11.43$ - $\cot(2^\circ) = \frac{1}{\tan(2^\circ)} \approx \frac{1}{0.03492} \approx 28.65$ So, $$d_w = 90 \times 11.43 = 1028.7 \text{ ft}$$ $$d_s = 90 \times 28.65 = 2578.5 \text{ ft}$$ 9. **Step 5: Calculate distance between fires:** $$D = \sqrt{1028.7^2 + 2578.5^2} = \sqrt{1057969 + 6654982} = \sqrt{7712951} \approx 2777 \text{ ft}$$ **Final answer:** The fires are approximately 2777 feet apart. --- **How to find angle of depression:** - The angle of depression is measured from the horizontal line at the observer's eye level down to the object. - If you know the height and horizontal distance, you can find it using: $$\theta = \arctan\left(\frac{\text{height}}{\text{horizontal distance}}\right)$$ - Conversely, if you know the angle of depression and height, you can find horizontal distance using: $$\text{horizontal distance} = \text{height} \times \cot(\theta)$$