3.10 HW Problems

Table of Contents:

Introduction to Related Rates

• What do we do when two+ related quantities are changing?
  • the problem of determining how the rate of change of one of them affects the rate of change of others is called a related rates problem

Related Rates Equations

• Let’s say we are pumping a spherical balloon.

  • BOTH the volume and radius are increasing overtime

  • ($V$ = volume and $r$ = the radius of the balloon at an instant time)

    $$ V=\frac{4}{3}\pi r^3 $$

  • using the Chain Rule, we differentiate both sides with respect to t to find the equation relating to the rates of change of V and r,

  $$ \frac{dV}{dt}=\frac{dV}{dr}\frac{dr}{dt}=4\pi r^2\frac{dr}{dt} $$

Example

Water runs into a conical tank at the rate of $9\,\, ft^3/min$. The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is teh water level rising when the water is 6 ft deep?

Notice the similar triangles.

• We assume that V, x, and y are differentiable functions of t. We are asked for $\frac{dy}{dt}$when

$$ y=6\,ft \enspace \text{and} \enspace \frac{dV}{dt}=9\,ft^3/min $$

• The water forms a cone with volume:

$$ V=\frac{1}{3}\pi x^2y $$

• This equation involves x as well as V and y. Because no information is given about x and $\frac{dx}{dt}$ at the time in question, we need to eliminate x.
  • the similar triangles give us a good way to express x in terms of y

$$ \frac{x}{y}=\frac{5}{10} \enspace \text{or} \enspace x=\frac{y}{2} $$

• Therefore we find,