Related rates
Rates that depend on each other
A ladder leans against a wall, and its foot slides away at a steady 1 metre per second. Does the top slide down at a steady speed too?
Play each scene below and watch the graph under it. One rate is held constant. The other changes, because the two quantities are tied together by an equation. Predict the shape of the graph before you press Play.
A method that works every time
- 1. Draw and name
Sketch the situation and give a letter to every quantity that changes. Note which rate you know and which one you want.
- 2. Relate
Find an equation linking the quantities, such as Pythagoras, an area formula, a volume formula, or similar triangles. Do not substitute changing values yet.
- 3. Differentiate
Differentiate both sides with respect to time t. Every changing quantity gets a chain-rule factor: \frac{d}{dt}x^2=2x\frac{dx}{dt}.
- 4. Substitute and solve
Only now plug in the values at the moment you care about, and solve for the unknown rate.
For the ladder, x^2+y^2=100 gives 2x\frac{dx}{dt}+2y\frac{dy}{dt}=0, so \frac{dy}{dt}=-\frac{x}{y}\frac{dx}{dt}. As the foot moves out, x grows and y shrinks. What does that do to the speed of the top?
Solve it step by step
Work each problem through the four steps. Answers can use pi, such as 20pi or 1/pi. The score counts steps you answer correctly on the first try.
Common mistakes
- Substituting too early
If you put x=6 into the equation before differentiating, x becomes a constant and its rate disappears.
- Ignoring the sign
A negative rate means the quantity is decreasing. The top of the ladder has \frac{dy}{dt}<0 because it is moving down.
- Keeping too many variables
In the cone, use similar triangles to write r in terms of h before differentiating, so only one rate is unknown.