1. **Problem 1: Angle of Depression from Hot Air Balloon** We are given a hot air balloon 40 ft above the ground and 70 ft away horizontally from a farm. We need to find the angle of depression from the balloon to the farm. 2. **Formula and Explanation:** The angle of depression corresponds to the angle between the horizontal line from the balloon and the line of sight to the farm. We can model this as a right triangle where: - Opposite side = height difference = 40 ft - Adjacent side = horizontal distance = 70 ft We use the tangent function: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{40}{70}$$ 3. **Calculation:** Calculate the angle $\theta$: $$\theta = \tan^{-1}\left(\frac{40}{70}\right) = \tan^{-1}(0.5714)$$ Using a calculator: $$\theta \approx 29.74^\circ$$ 4. **Answer:** The angle of depression is approximately **29.74 degrees**. 5. **Problem 2: Distance Between Airports A and B** An airplane travels 50 miles North 35° West (N35°W), then 70 miles South 20° West (S20°W). We want to find the straight-line distance between airports A and B. 6. **Understanding Directions and Components:** - N35°W means 35° west of north. - S20°W means 20° west of south. We break each leg into components using trigonometry, assuming north is positive y-axis and east is positive x-axis. For the first leg (50 miles N35°W): - $x_1 = -50 \times \sin(35^\circ)$ (west is negative x) - $y_1 = 50 \times \cos(35^\circ)$ For the second leg (70 miles S20°W): - $x_2 = -70 \times \sin(20^\circ)$ - $y_2 = -70 \times \cos(20^\circ)$ (south is negative y) 7. **Calculate components:** $$x_1 = -50 \times 0.574 = -28.7$$ $$y_1 = 50 \times 0.819 = 40.95$$ $$x_2 = -70 \times 0.342 = -23.94$$ $$y_2 = -70 \times 0.940 = -65.8$$ 8. **Total displacement components:** $$x = x_1 + x_2 = -28.7 - 23.94 = -52.64$$ $$y = y_1 + y_2 = 40.95 - 65.8 = -24.85$$ 9. **Distance between airports:** Use Pythagorean theorem: $$d = \sqrt{x^2 + y^2} = \sqrt{(-52.64)^2 + (-24.85)^2}$$ Calculate: $$d = \sqrt{2771.5 + 617.6} = \sqrt{3389.1} \approx 58.23$$ 10. **Answer:** The distance between airports A and B is approximately **58.23 miles**.