Related Rates Studio
Quantities that are tied together change together. Watch a scene move, see the chain rule link the rates, and check the answer against the motion itself. Then practice the full round table: picture, equation, derivative, substitution and a sentence in context.
Right nowUnits of each quantity
Increasing or decreasing?
Rates over time
Try this
Picture it
Draw your own picture first. Then compare it with this one, frozen at the instant in the problem.
Arrows show which way quantities in the problem are changing. Colors: green is a known rate, orange is the rate to find, blue is found along the way.
Round table
Every related rates problem follows the same six moves.
- Draw a pictureLabel every quantity that changes with a letter. Put numbers only on what stays fixed.
- Write the equationRelate the letters with geometry (Pythagorean theorem, area or volume, similar triangles), or use the equation the problem gives you.
- Differentiate with respect to tEvery changing letter gets its chain-rule factor, such as dx/dt.
- Substitute at the instantOnly now put in the numbers. Use the sign of each rate: decreasing means negative.
- SolveSolve for the unknown rate.
- Interpret with unitsSay what is changing, which way, how fast, and in what units.
Every problem is generated fresh, and every answer key is computed in code and checked with a computer algebra system. Scores stay in this browser only. Non-calculator. Topics 4.4 Introduction to Related Rates and 4.5 Solving Related Rates Problems.