Definition: Related rates problems connect how fast several changing quantities move over time, using an equation that links them and differentiating it with respect to time.
What is a related rates problem?
Several quantities change with time and a geometric or physical equation ties them together. Differentiate that equation with respect to t, then substitute the known values.
What are the steps?
- Draw a picture and name the variables.
- Write an equation connecting them.
- Differentiate with respect to time.
- Substitute numbers only after differentiating.
Worked example: a 13-foot ladder leans on a wall. Its base slides away at 3 ft/s. How fast does the top fall when the base is 5 ft from the wall? Use x2 + y2 = 169. When x = 5, y = 12. Differentiate: 2x(dx/dt) + 2y(dy/dt) = 0, so 5(3) + 12(dy/dt) = 0 and dy/dt = -1.25 ft/s.
What is the common error?
Plugging in numbers before differentiating, which turns changing quantities into constants. Mathbot keeps letters until the derivative is done.