Related rates from models to interpretation

Understand the idea

Name every time-dependent quantity and write a geometric or physical relationship valid throughout the motion. Differentiate with respect to time before substituting the instantaneous values. Keep signs on given rates and solve for the requested rate. A diagram helps distinguish changing distances from fixed dimensions.

\frac d{dt}(x^2+y^2)=2x\frac{dx}{dt}+2y\frac{dy}{dt}

Worked example

A 10-foot ladder has its foot moving away from a wall at $\displaystyle 2$ ft/s. How fast is the top moving when the foot is $\displaystyle 6$ feet from the wall?

Let

\displaystyle x

be the foot distance and

\displaystyle y

the height. From

\displaystyle x^2+y^2=100

,

\displaystyle y=8

at this instant. Differentiate:

\displaystyle 2x x'+2y y'=0

. Thus

\displaystyle y'=-(6)(2)/8=-3/2

ft/s, so the top moves downward.

Try it yourself

A circle radius increases at $\displaystyle 3$ cm/s. Find the area rate when $\displaystyle r=2$ cm.

Show solution and reasoning

$\displaystyle A=\pi r^2$ gives $\displaystyle dA/dt=2\pi r\,dr/dt=12\pi$ square centimeters per second.