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Related Rates

Calculus Related Rates Problem Solving Strategy

We will use the steps outlined below to solve each Related Rates problem on this site, step-by-step, every single time. We hope that this will help you see the strategy we’re using so you can learn it too, and then be able to apply it to all of your problems, especially those on your exams. That’s as opposed to learning just how to solve a particular problem on your homework, say, since you may well never see that specific problem again.

We developed this strategy in our blog post 4 Steps to Solve Any Related Rates Problem. (Link will open in a new tab.) You might find it helpful to read that post before proceeding if you haven’t already.

  1. Draw a picture of the physical situation.
  2. Write an equation that relates the quantities of interest.
    1. Be sure to label as a variable any value that changes as the situation progresses; don’t substitute a number for it yet.
    2. To develop your equation, you will probably use:
      • a simple geometric fact (like the relation between a sphere’s volume and its radius, or the relation between the volume of a cylinder and its height); or
      • a trigonometric function (like $\tan{\theta}$ = opposite/adjacent); or
      • similar triangles; or
      • the Pythagorean theorem.
  3. Take the derivative with respect to time of both sides of your equation. Remember the chain rule.
  4. Solve for the quantity you’re after.

On our Related Rates page, we solve an example of each of the approaches listed in 2.B above:

i. Geometric fact. Typical problems: A circle’s radius changes, a snowball melts, a rectangle’s height and/or width changes.

ii. Trig function. Typical problems: A searchlight rotates, a rocket takes off, a kite travels horizontally.

iii. Similar triangles. Water fills a cone or trough, sand falls onto a conical pile, person walks away from a light pole that casts a shadow.

iv. Pythagorean theorem. Typical problems: Cars/ships/joggers move along 90 degree paths, baseball players run along the diamond, boat is pulled toward a dock.

To see an example of each type, please visit our Related Rates page.

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