1. Problem 1: An airplane is approaching a runway with an angle of depression of 3° and altitude 500 feet. We want to find the straight line distance to the runway. 2. Use the right triangle trigonometry formula: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \). 3. Here, \( \theta = 3^\circ \), adjacent side = altitude = 500 feet, hypotenuse = distance to runway (unknown). 4. Set up the equation: \( \cos(3^\circ) = \frac{500}{d} \). 5. Solve for \( d \): \( d = \frac{500}{\cos(3^\circ)} \). 6. Calculate \( \cos(3^\circ) \approx 0.9986 \), so \( d \approx \frac{500}{0.9986} = 500.7 \) feet. --- 7. Problem 2: A model rocket is launched. You stand 100 feet from the pad and measure an angle of elevation of 77°. Find the highest point's height. 8. Use \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). 9. Here, \( \theta = 77^\circ \), adjacent = 100 feet, opposite = height (unknown). 10. Set up: \( \tan(77^\circ) = \frac{h}{100} \). 11. Solve for \( h \): \( h = 100 \times \tan(77^\circ) \). 12. Calculate \( \tan(77^\circ) \approx 4.3315 \), so \( h \approx 433.15 \) feet. --- 13. Problem 3: A wheelchair ramp must have an angle of elevation \( \leq 4.78^\circ \). The ramp length is 16 feet, vertical rise is 14 inches. 14. Convert 14 inches to feet: \( \frac{14}{12} = 1.1667 \) feet. 15. Use \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). 16. Calculate actual angle: \( \theta = \arcsin\left(\frac{1.1667}{16}\right) \). 17. Compute \( \frac{1.1667}{16} = 0.0729 \), so \( \theta = \arcsin(0.0729) \approx 4.18^\circ \). 18. Since \( 4.18^\circ < 4.78^\circ \), the ramp meets federal standards. Final answers: - Distance to runway: \( \boxed{500.7} \) feet. - Rocket highest point: \( \boxed{433.15} \) feet. - Ramp angle: \( 4.18^\circ \), meets standard: \( \boxed{\text{Yes}} \).