Time Rates

Rate of Change of Volume of Sand in Conical Shape

Problem
A conveyor is dispersing sands which forms into a conical pile whose height is approximately 4/3 of its base radius. Determine how fast the volume of the conical sand is changing when the radius of the base is 3 feet, if the rate of change of the radius is 3 inches per minute.

A.   2π ft/min C.   3π ft/min
B.   4π ft/min D.   5π ft/min

 

Velocity of Separation: How fast is the distance between two cars changing?

Problem
A Toyota Land Cruiser drives east from point A at 30 kph. Another car, Ford Expedition, starting from B at the same time, drives S30°W toward A at 60 kph. B is 30 km from A. How fast in kph is the distance between two cars changing after 30 minutes? Hint: Use the Cosine Law.

A.   70 kph C.   55 kph
B.   80 kph D.   60 kph

 

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.

49-51 Ship sailing east and turned N 30d E

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.

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.

45 - Angle of elevation of the the kite's cord

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.
 

44 - Angle of elevation of the rope tied to a rowboat on shore

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.
 

40 - Base angle of a growing right triangle

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°.