52-53 Two cars traveling from the same point but going to different directions

Problem 52
A car drives south at 20 mi/hr. Another car, starting from the same point at the same time and traveling 40 mi/hr, goes east for 30 minutes then turns north. Find the rate of rotation of the line joining the cars (a) 1 hour after the start; (b) at the time the second car makes its turn.

Problem 53
Prove that the results in Problem 52 are independent of the speed of the cars, if the second car travels twice as fast as the first car.

46-48 Rate of rotation of the searchlight

Problem 46
A ship, moving at 8 mi/hr, sails east for 2 hr, then turns N 30° W. A searchlight, placed at the starting point, follows the ship. Find how fast the light is rotating, (a) 3 hr after the start; (b) just after the turn.

Problem 47
In Problem 46, find when the light rotates most rapidly.

Problem 48
Prove that the result in Problem 46 and Problem 47 are independent of the speed of the ship.