# Time Rates

**Problem**

Gas is escaping from a spherical balloon at a constant rate of 2 ft^{3}/min. How fast, in ft^{2}/min, is the outer surface area of the balloon shrinking when the radius is 12 ft?

A. 1/2 | C. 1/3 |

B. 1/5 | D. 1/4 |

**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 |

**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

## 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°.

## 39 - Rate of increase of angle of elevation of the line of sight

**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?