4.6 - Related Rates

Essentially the same as 4.4 - Optimization, but this time, there are two rates, and so now you have to do Implicit Differentiation.

What is a related rate?

Rates that are related :D

• For example, the surface area and volume of a sphere, or the brightness of a star from Earth and its distance.
• Related rates are expressed in terms of formulas – the area of a circle and its radius are related; this relation comes from the formula $A=\pi r^2$.

Solving a related rate problem

• Essentially, what $\frac{d?}{dt}$ do we already know, and what $\frac{d?}{dt}$ do we need?

Example problem

Life of Pi

A boat is pulled toward a dock by a rope from the bow using a tool that is $6$ feet above the bow. If the rope is pulled in at $2$ ft/sec, how fact is the boat approaching the dock when $10$ feet of rope are out?

We use the Pythagorean theorem to develop our model: we need to relate $x$ and $r$ in some way to begin. We can say that $x^2+36=r^2$. We then take the derivative of this formula using implicit differentiation, getting $2x\frac{dx}{dt}+0=2r\frac{dr}{dt}$. We know that $x$ and $r$ are values of a right triangle, so we know that $x=8$ and $r=10$. Given that we also know $\frac{dr}{dt}$, we finally solve our equation for $\frac{dx}{dt}$, getting that $\frac{dx}{dt}=2.5$ ft/sec.