Related Rates

Imagine a cop with a radar gun, waiting on the side of the street.

You pass by in a car. The cop wants to know if you're speeding or not.

Two quantities are changing with respect to time: the distance between you and the cop (call it $z$), and the distance between you and the point straight ahead of you perpendicular to the cop's line of vision to the road (call this distance $x$).

It is easier to measure the rate of change of the distance $z$; the radar gun can measure this.

However, the cop wants to know your real speed, or the rate of change or $x$.

Related rates problems are typically in the above setup. The rate of change of one quantity is easier to measure, and that rate will also tell us something about the harder to measure rate of change.

In general, two quantities $x$ and $y$ will depend on a third quantity $t$. Given $dx/dt$, can we find $dy/dt$?

Solving related rates problems
  1. Assign a variable to each quantity that are functions of time. Draw a diagram to help you if needed.
  2. Write out the given information and the required rate in terms of derivatives.
  3. If no equation is given, write an equation that relates the various quantities of the problem.
  4. Implicitly differentiate both sides of the equation with respect to time.
  5. If necessary, use the information given to eliminate variables from the original equation with substitution.
  6. Solve for the unknown rate.
Air is being pumped into a spherical balloon so that it volume increases at a rate of 100 cm$^3$/s. How fast is the radius of the balloon increasing when the diameter is 50 cm?
When the diameter of a spherical tumor is 16mm it is growing at a rate of 0.4 mm a day. How fast is the volume of the tumor changing at that time?
At a distance of 4000 feet from the launch site, a spectator is observing a rocket being launched. If the rocket lifts off vertically and is rising at a speed of 600 feet/second when it is at an altitude of 3000 feet, how fast is the distance between the rocket and the spectator changing at that instant?
A 10 foot ladder is leaning against a vertical wall. If the bottom of the ladder is sliding away from the wall at a rate of 1 foot per second, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet away from the wall?