Two boats leave a port at the same time, one traveling west at 20 mi/hr and the other traveling southwest ( 45° south of west) at 15 mi/hr. After 30 minutes, how far apart are the boats and at what rate is the distance between them changing? (Hint: Use the Law of Cosines.)

Verified step by step guidance

1. Convert the speeds of the boats from miles per hour to miles per minute since the time given is in minutes. The westbound boat travels at 20 mi/hr, which is approximately 20/60 = 1/3 mi/min, and the southwest boat travels at 15 mi/hr, which is approximately 15/60 = 1/4 mi/min.
2. Determine the distance each boat travels in 30 minutes. For the westbound boat, the distance is (1/3 mi/min) * 30 min = 10 miles. For the southwest boat, the distance is (1/4 mi/min) * 30 min = 7.5 miles.
3. Use the Law of Cosines to find the distance between the two boats after 30 minutes. The angle between the two boats is 135° (90° + 45°). The Law of Cosines states that c² = a² + b² - 2ab * cos(θ), where a and b are the distances traveled by the boats, and θ is the angle between them.
4. Substitute the values into the Law of Cosines formula: c² = (10 miles)² + (7.5 miles)² - 2 * (10 miles) * (7.5 miles) * cos(135°). Calculate the cosine of 135° and simplify the expression to find c².
5. To find the rate at which the distance between the boats is changing, differentiate the Law of Cosines with respect to time, applying the chain rule, and substitute the known values of the distances and their rates of change.