AP Calculus BC / Unit 4: Contextual Applications of Differentiation
4.5 Solving Related Rates Problems
Notes
Steps: draw a picture, name variables, write an equation linking them, differentiate with respect to $t$, then substitute the known values and solve.
- Use geometry (similar triangles, Pythagorean theorem) to get the equation.
- Check signs: a decreasing quantity has a negative rate.
Practice
A circle's radius grows at 2 cm/s. How fast is the area growing when $r=5$?
$\displaystyle \frac{dA}{dt}=2\pi(5)(2)$.
A 10 ft ladder slides; its base moves away at 1 ft/s. How fast is the top sliding down when the base is 6 ft out?
Top is 8 ft high: $6(1)+8y'=0$, so $\displaystyle y'=-\frac{3}{4}$.
A balloon's volume grows at $100$ cm³/s. Its radius rate when $r=5$ is ($\displaystyle V=\frac{4}{3}\pi r^3$):
$100=4\pi(25)r'$, so $\displaystyle r'=\frac{1}{\pi}$.
Ready for more? Take the Unit 4 test.