Related rates problems
When two quantities change together, differentiating their relationship with respect to time connects their rates.
Strategy
Write an equation relating the variables themselves, using only variables at this stage, not the given rates. Differentiate both sides with respect to time t, applying the chain rule so every term picks up its own rate. Finally substitute the specific values and the known rates, then solve for the unknown rate.
\frac{d}{dt}[x^2] = 2x \frac{dx}{dt}Worked example
A 10-foot ladder leans against a wall with its base sliding away at 1 foot per second. Let x be the base distance and y the height, so x^2 + y^2 = 100. Differentiating gives 2x (dx/dt) + 2y (dy/dt) = 0. When x = 6, y = 8, so 6(1) + 8 (dy/dt) = 0 and the top slides down at 0.75 feet per second.
6(1) + 8 \frac{dy}{dt} = 0 \implies \frac{dy}{dt} = -0.75Reading the sign
A negative rate means the quantity is decreasing, not that a mistake was made. In the ladder problem the height y shrinks as the base x grows, so dy/dt comes out negative. Always answer the question asked: a speed is the absolute value of the rate.
\frac{dy}{dt} < 0 \text{ means } y \text{ is decreasing}Flashcard
Pythagorean link between ladder base and height
Check your understanding
A spherical balloon inflates so its radius grows at 2 cm/s. How fast is the volume increasing when the radius is 3 cm?