Learning Outcomes

  1. Express changing quantities in terms of derivatives.
  2. Use the chain rule to find the rate of change of one quantity that depends on the rate of change of other quantities.


  • Chapter 4.1  Related Rates  

Textbook Assignment

  • p. 350:    1, 5, 10, 17, 20, 25, 29

WeBWork Assignment

  • Application-Related Rates

Exit problems of the session 

  1. The radius of sphere increases at a rate of 2m/s. Find the rate at which the volume increases when the volume is 36$\pi$.

  2. A vertical cylinder is leaking water at a rate of 1ft$^3$/sec. If the cylinder has a height of 10 ft and a radius of 1 ft, at what rate is the height of the water changing when the height is 6ft?

  3. Sand pours from a chute and forms a conical pile whose height is always equal to its base diameter. The height of the pile increases at a rate of 5 ft/hr. Find the rate of change of the volume of the sand in the conical pile, when the height of the pile is 4 ft.

  4. A 13-ft ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2ft/sec, how fast is the bottom moving along the ground when the bottom of the ladder is 5 ft from the wall?


Key Concepts

  • Solving a related-rates problem:
    1. To solve a related rates problem, first draw a picture that illustrates the relationship between the two or more related quantities that are changing with respect to time. Assign symbols to all variables involved in the problem. 
    2. In terms of the variables, state the information given and the rate to be found.
    3. Find an equation relating the variables.
    4. Use differentiation, applying the chain rule as necessary, to find an equation that relates the derivatives.
    5. Substitute all known values into the equation from step 4, then solve for the unknown rate of change.
  • Useful formulas:
    circumference of a circle:  $C=2\pi r$     area of a circle:   $A=\pi r^2$
    surface area of a sphere:  $S=4\pi r^2$    volume of a sphere:  $V=\frac{4}{3}\pi r^3$
    volume of a cylinder:  $V=\pi r^2 h$
    volume of a cone:  $V=\frac{1}{3}\pi r^2 h$

Videos and Practice Problems of Selected Topics

  1. $\rhd$ Approaching cars (6:52) As two cars approach the same intersection from different roads, how does the rate of change of the distance between them change?
  2. $\rhd$ Falling ladder (5:48) You’re on a ladder.  The bottom of the ladder starts slipping away from the wall. Analyze the rate of change.
  3. $\rhd$ Water pouring into a cone (11:31) As you pour water into a cone, how does the rate of change of the depth of the water relate to the rate of change in volume?
  4. * Practice: Related rates problems using the Pythagorean Theorem. (4 problems)
  5. * Practice: Related rates advanced problems. (4 problems)

STEM Applications