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.
 

43 - 45 Solved problems in maxima and minima

Problem 43
A ship lies 6 miles from shore, and opposite a point 10 miles farther along the shore another ship lies 18 miles offshore. A boat from the first ship is to land a passenger and then proceed to the other ship. What is the least distance the boat can travel?
 

Problem 44
Two posts, one 8 feet high and the other 12 feet high, stand 15 ft apart. They are to be supported by wires attached to a single stake at ground level. The wires running to the tops of the posts. Where should the stake be placed, to use the least amount of wire?
 

Problem 45
A ray of light travels, as in Fig. 39, from A to B via the point P on the mirror CD. Prove that the length (AP + PB) will be a minimum if and only if α = β.