Calculus / Unit 5: Applications of Derivatives I
5.4 Related Rates Examples
Notes
Common setups: Pythagorean theorem (ladders, distances), volume formulas (spheres, cubes, cones), area formulas.
- Ladder: $x^2 + y^2 = L^2$ gives $\displaystyle x\frac{dx}{dt} + y\frac{dy}{dt} = 0$ because $L$ is constant.
- Sphere: $\displaystyle V = \frac{4}{3}\pi r^3$ gives $\displaystyle \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$.
- Cube: $V = s^3$ gives $\displaystyle \frac{dV}{dt} = 3s^2 \frac{ds}{dt}$.
Practice
A 13 ft ladder leans on a wall. The base is 5 ft from the wall and slides away at 2 ft/s. How fast is the top sliding down the wall?
The height is 12. From $\displaystyle 5(2) + 12\frac{dy}{dt} = 0$, $\displaystyle \frac{dy}{dt} = -\frac{5}{6}$, so the top moves down at $\displaystyle \frac{5}{6}$ ft/s.
A spherical balloon's radius grows at 0.5 cm/s. How fast is the volume growing when $r = 3$ cm?
$\displaystyle \frac{dV}{dt} = 4\pi r^2\frac{dr}{dt} = 4\pi(9)(0.5) = 18\pi$.
A cube's edge grows at 0.5 cm/s. How fast is the volume growing when the edge is 2 cm?
$\displaystyle \frac{dV}{dt} = 3s^2\frac{ds}{dt} = 3(4)(0.5) = 6$.
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