A DC-9 aircraft leaves an airport from a runway whose bearing is N39 \( { }^{\circ} \mathrm{E} \). After flying for \( \frac{1}{4} \) mile, the pilot requests permission to turn \( 90^{\circ} \) and head toward the southeast. The permission is granted. After the airplane goes 1 mile in this direction, what bearing should the control tower use to locate the aircraft? What is the bearing that the control tower should use to locate the aircraft? (Do not round until the final answer. Then round to the nearest tenth as needed.) -

To determine the bearing that the control tower should use to locate the aircraft, we'll break down the flight path into two segments and use trigonometry to find the final position relative to the airport. ### Step 1: Initial Flight Segment - **Bearing:** N39°E - **Distance:** \( \frac{1}{4} \) mile Using a coordinate system where: - **North** is the positive \( y \)-axis. - **East** is the positive \( x \)-axis. The angle from the east (\( x \)-axis) is: \[ \theta_1 = 90^\circ - 39^\circ = 51^\circ \] **Displacement Components:** \[ \begin{align*} \Delta x_1 &= \frac{1}{4} \times \cos(51^\circ) \approx 0.1573 \text{ miles} \\ \Delta y_1 &= \frac{1}{4} \times \sin(51^\circ) \approx 0.1943 \text{ miles} \end{align*} \] ### Step 2: Turn and Second Flight Segment - **Turn:** 90° to the left (to head southeast) - **New Bearing Angle:** \( \theta_2 = 51^\circ + 90^\circ = 141^\circ \) - **Distance:** 1 mile **Displacement Components:** \[ \begin{align*} \Delta x_2 &= 1 \times \cos(141^\circ) \approx -0.7771 \text{ miles} \\ \Delta y_2 &= 1 \times \sin(141^\circ) \approx 0.6293 \text{ miles} \end{align*} \] ### Step 3: Total Displacement \[ \begin{align*} \Delta x_{\text{total}} &= \Delta x_1 + \Delta x_2 \approx 0.1573 - 0.7771 = -0.6198 \text{ miles} \\ \Delta y_{\text{total}} &= \Delta y_1 + \Delta y_2 \approx 0.1943 + 0.6293 = 0.8236 \text{ miles} \end{align*} \] ### Step 4: Calculating the Bearing The negative \( x \)-component indicates west, and the positive \( y \)-component indicates north. To find the bearing: 1. Calculate the angle \( \phi \) from the north: \[ \phi = \tan^{-1}\left(\frac{|\Delta x_{\text{total}}|}{\Delta y_{\text{total}}}\right) = \tan^{-1}\left(\frac{0.6198}{0.8236}\right) \approx 36.96^\circ \] 2. Round to the nearest tenth: \[ \phi \approx 37.0^\circ \] **Final Bearing:** \( \text{N}37.0^\circ\text{W} \) ### Conclusion The control tower should use a bearing of **N37.0°W** to locate the aircraft. **Answer:** After rounding, the tower should use a bearing of N 37.0 ° W to locate the aircraft.