# Related rates

For each of the following, it may help to find some equation that relates the information in the problem, then use implicit differentiation to find the related rates.

1. A 13-foot ladder is leaning against a house when its base starts to slide away. By the time the base is 12 feet from the house, the base is moving at the rate of 6 feet per second. What is the rate the ladder sliding down the wall at this time?

2. Assume that the infected area of an injury is circular and the radius of the injury is growing at a rate of 0.5 mm per hour. At what rate is the area of the injury increasing when the radius is 6mm?

3. An airplane is flying away from an airport at an altitude of 4 miles. At a specific time, the radar at the airport detects the diagonal distance between the plane and the airport is changing at a rate of 240 miles per hour and the horizontal distance is 40 miles. If the plane flies at a constant altitude, what is the horizontal velocity of the airplane at this time?

## Challenges

4. A conical tank is twice as wide as it is high. Water is draining from the tank at the rate of 6 cubic feet per minute. How fast is the depth of water in the tank declining when it is 4 feet deep?

5. A balloon leaves the ground 500 feet from an observer and rises vertically at a rate of 140 feet per minute. At what rate is the angle of inclination from the observer increasing the instant the balloon is 500 feet off the ground?

6. The volume of a cube is increasing at a rate of 0.1 cubic cm per second when the side length is 10 cm. What is rate of the surface area at this same time?