A plane leaves airport A and travels 580 miles to airport B on a bearing of \( \mathrm{N} 34^{\circ} \mathrm{E} \). The plane later leaves airport B and travels to airport C 400 miles away on a bearing of \( \mathrm{S} 74^{\circ} \mathrm{E} \). Find the distance from airport A to airport C to the nearest tenth of a mile.

To find the distance from airport A to airport C, we can employ the Law of Cosines after determining the coordinates of each airport. First, we convert the bearings into standard angles: - From A to B: \( 34^\circ \) from North corresponds to \( 90^\circ - 34^\circ = 56^\circ \) from the horizontal. - From B to C: \( 74^\circ \) from South corresponds to \( 74^\circ + 90^\circ = 164^\circ \). Next, we can find the coordinates of airports B and C, starting from A (which we place at the origin [0, 0]): - Airport B: - \( x_B = 580 \cos(34^\circ) \) - \( y_B = 580 \sin(34^\circ) \) - Airport C: The angle needs to be adjusted for B's coordinates: - From B, C's coordinates relative to B are: - \( x_C = 400 \cos(164^\circ) \) - \( y_C = 400 \sin(164^\circ) \) - Therefore, C's coordinates relative to A become: - \( x_C = x_B + 400 \cos(164^\circ) \) - \( y_C = y_B + 400 \sin(164^\circ) \) Finally, we compute the distance \( d \) from A to C using the distance formula: \[ d = \sqrt{(x_C - x_A)^2 + (y_C - y_A)^2} \] Calculating the above will yield the distance from airport A to airport C. Let's do the calculations quickly: 1. \( x_B \approx 580 \cdot 0.829 \approx 480.82 \) 2. \( y_B \approx 580 \cdot 0.515 \approx 298.7 \) 3. \( x_C \approx 400 \cdot -0.439 \approx -175.6 \) (east decreases x) 4. \( y_C \approx 400 \cdot 0.900 \approx 360 \) Now we find \( x_C \) and \( y_C \): - \( x_C \approx 480.82 - 175.6 \approx 305.22 \) - \( y_C \approx 298.7 + 360 \approx 658.7 \) Finally, using these coordinates in the distance formula: \[ d = \sqrt{(305.22 - 0)^2 + (658.7 - 0)^2} \] \[ d \approx \sqrt{(305.22)^2 + (658.7)^2} \] \[ d \approx 733.1 \text{ miles} \] So, the distance from airport A to airport C is approximately 733.1 miles.