Questions: A ship travels 52 km on a bearing of 20 degrees, and then travels on a bearing of 110 degrees for 113 km. Find the distance from the starting point to the end of the trip, to the nearest kilometer.

Solution Steps

To solve this problem, we can use the law of cosines. First, we need to determine the angle between the two paths of the ship. Since bearings are measured clockwise from the north, the angle between the two paths is the difference between the two bearings. Once we have this angle, we can apply the law of cosines to find the distance from the starting point to the end of the trip.

The ship travels on a bearing of \(20^\circ\) initially and then changes to a bearing of \(110^\circ\). The angle between these two paths is the difference between the bearings: \[ \text{Angle between paths} = 110^\circ - 20^\circ = 90^\circ \]

To use the law of cosines, we need the angle in radians. The conversion from degrees to radians is given by: \[ \text{Angle in radians} = \frac{90 \times \pi}{180} = \frac{\pi}{2} \approx 1.5708 \]

The law of cosines states: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \] where \(a = 52\) km, \(b = 113\) km, and \(C = \frac{\pi}{2}\). Substituting these values, we have: \[ c^2 = 52^2 + 113^2 - 2 \times 52 \times 113 \times \cos\left(\frac{\pi}{2}\right) \] Since \(\cos\left(\frac{\pi}{2}\right) = 0\), the equation simplifies to: \[ c^2 = 52^2 + 113^2 \] \[ c = \sqrt{52^2 + 113^2} = \sqrt{2704 + 12769} = \sqrt{15473} \approx 124.3905 \]

The distance from the starting point to the end of the trip is rounded to the nearest kilometer: \[ \text{Distance} \approx 124 \text{ km} \]

Final Answer