inverse trigonometric function

Problem 55
The lower edge of the picture is a ft, the upper edge is b ft, above the eye of an observer. At what horizontal distance should he stand, if the vertical angle subtended by the picture is to be greatest?
 

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 49
A ship, moving 10 mi/hr, sails east for 2 hours, then turns N 30° E. A searchlight, placed at the starting point, follows the ship. Find how fast the light is rotating (a) 4 hours after the start; (b) just after the turn.

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 45
A kite is 60 ft high with 100 ft of cord out. If the kite is moving horizontally 4 mi/hr directly away from the boy flying it, find the rate of change of the angle of elevation of the cord.
 

Problem 44
A rowboat is pushed off from a beach at 8 ft/sec. A man on shore holds a rope, tied to the boat, at a height of 4 ft. Find how fast the angle of elevation of the rope is decreasing, after 1 sec.
 

Problem 40
The base of a right triangle grows 2 ft/sec, the altitude grows 4 ft/sec. If the base and altitude are originally 10 ft and 6 ft, respectively, find the time rate of change of the base angle, when the angle is 45°.
 

Problem 39
A balloon, leaving the ground 60 ft from an observer, rises 10 ft/sec. How fast is the angle of elevation of the line of sight increasing, after 8 seconds?
 

Problem 38
A ship, moving 8 mi/hr, sails north for 30 min, then turns east. If a searchlight at the point of departure follows the ship, how fast is the light rotating 2 hr after the start.
 

Problem 37
A ladder 15 ft long leans against a vertical wall. If the top slides down at 2 ft/sec, how fast is the angle of elevation of the ladder decreasing, when the lower end is 12 ft from the wall?