Section 5.4 Related Rates
In some applications, two quantities that are related to each other are also dependant on time. Suppose that \(y = f(x)\text{,}\) and \(x\) varies with time \(t\text{.}\) Then, because \(y\) depends on \(x\text{,}\) \(y\) will also vary with time. That is, if \(y\) is a function of \(x\text{,}\) and \(x\) is a function of \(t\text{,}\) then \(y\) is a function of \(t\) (this is just function composition). Then, differentiating \(y\) with respect to \(t\text{,}\) the chain rule says that,
\begin{equation*}
\frac{dy}{dt} = \underbrace{\frac{dy}{dx}}_{\text{or } f'(x)} \cdot \frac{dx}{dt}
\end{equation*}
Then, the two rates of change \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\) are related through this equation. In general, if two related quantities \(A\) and \(B\) vary over time \(t\text{,}\) then the derivatives,
\begin{equation*}
\frac{dA}{dt} \quad \text{and} \quad \frac{dB}{dt}
\end{equation*}
are called related rates. Quite literally, they are rates which are related to each other. We can determine these derivatives by using the equation that relates \(A\) and \(B\text{,}\) and differentiating both sides implicitly with respect to time \(t\text{.}\)
The revenue of a company \(R\) is a function of the number of employees \(n\) is has, and both \(R\) and \(n\) also change over time.
Subsection 5.4.1 Solving Related Rates Problems
- Determine the variables in the problem, and the relationships between them. Determine what values and rates of change are given, and which are to be determined.
- Sketch the problem, labelling constant values, and variables.
- Determine one or more equations which relate the variables and quantities involved. In particular, an equation which relates the quantity whose rate is wanted to the quantity whose rate is given.
- Differentiate both sides of the equation implicitly with respect to time \(t\text{,}\) regarding all variables as functions of time, and using the chain rule.
- Substitute given values into the equation, and solve the resulting equation for the desired quantity.
- Check that your answer seems reasonable. For example, ensure it has the correct sign.
Subsection 5.4.2 Commonly-Used Formulas in Related-Rate Applications
- The Pythagorean theorem, \(x^2 + y^2 = r^2\text{.}\)
- Trigonometric functions, \(\sin{\theta} = \frac{y}{r}, \cos{\theta} = \frac{x}{r}, \tan{\theta} = \frac{y}{x}\text{.}\)
- Similar triangles, the ratio of corresponding sides are equal.
- Area and perimeter of a circle, \(A = \pi r^2\) and \(P = 2\pi r\text{.}\)
- Volume of a cone, \(V = \frac{1}{3} \pi r^2 h\text{.}\)
- Volume of a cylinder, \(V = \pi r^2 h\text{.}\)
- Law of cosines, \(c^2 = a^2 + b^2 - 2ab \cos{C}\)
Subsection 5.4.3 Remark About Related Rates
In fact, related rates are almost entirely academic applications, in that it has no real applications to problems people actually consider in the real world.
