Related Rates

We have already seen how the Chain Rule can be used to differentiate a function implicitly. Another important use of the Chain Rule is to find the rates of change of two or more related variables that are changing with respect to time. For example:

Suppose x and y are both differentiable functions of t and are related by the equation below. Find when x = 1, given that = 2 when x = 1. Answer: 4

Step 1 – Draw a picture (if it is needed to understand the problem) Step 2 – Determine what rate you are looking for (this will always be a differential) Step 3 – Determine what rate (or rates) is given Step 4 – Write an equation which relates the variables in Steps 2 and 3. Equate these variables. Step 5 – Differentiate the equation in Step 4 with respect to whatever variable you are given. Use the Chain Rule when differentiating. Step 6 – Substitute in what variables are known and solve for the rate in Step 1.

A hot-air balloon rising straight up from a level field is tracked by a range finder 500 feet from the lift-off point. At the moment the range finder’s elevation angle is, the angle is increasing at the rate of 0.14 rad./min. How fast is the balloon rising at that moment? Answer: 140 feet/minute

A police car, approaching a right-angled intersection from the north, is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.6 miles north of the intersection and the car is 0.8 miles to the east, the police determine with radar that the distance between them and the car is increasing at 20 miles per hour. If the cruiser is moving at 60 miles per hour at the instant of measurement, what is the speed of the car? Answer: 70 miles per hour

Water runs into a conical tank at the rate of 9. The tank stands point down and has a height of 10 feet and a base radius of 5 feet. How fast is the water level rising when the water is 6 feet deep? Answer: 0.32 feet/minute

How fast does the water level drop when any cylindrical tank is drained at the rate of 3 liters per second? Answer:

Air is being pumped into a spherical balloon at a rate of 4.5 cubic feet per minute. Find the rate of change of the radius when the radius is 2 feet. Answer: feet/minute

A right angled triangle ABC has a fixed hypotenuse AC of length 10 cm, and side AB increases at 0.1 cm per second. At what rate is angle CAB increasing at the instant when the triangle is isosceles? Answer:

An airplane lies horizontally away from an observer at an altitude of 5000 m, with an air speed of 200 m/s. At what rate is its angle of elevation to the observer changing at the instant when the angle of elevation is 60 degrees? Answer:

The area of a variable rectangle remains constant at 100 centimeters squared. The length of the rectangle is decreasing at 1 cm per minute. At what rate is the width increasing at the instant when the rectangle is a square? Answer: 1 cm per minute

A stone is thrown into a lake and a circular ripple moves out at a constant speed of 1 m/s. Find the rate of which the circle’s area is increasing at the instant when t = 4 seconds. Answer:

Wheat runs from a hole in a silo at a constant rate and forms a conical heap whose base radius is three times the height. If after 1 minute, the height of the heap is 20 cm, find the rate at which the height is rising at this instant. Answer: