When two, or more, variables are related, the rate of change with respect to time will also be related. This relation could come from a geometric formula, like the area of a circle, or the Pythagorean Theorem, or it could be two variables that are related by the sine ratio from a right triangle. The key concept to related rates is to consider each variable as being an explicit function of time, \(t\).
Think of this example. A drop of oil is placed on water. Since oil and water do not mix, as time passes, this drop begins to form a circular shape that is growing in area as the oil continues to spread out. There are two variables at play here, the area of the circle, \(A\), and the radius of that circle, \(r\). Both of these variables depend on how much time has passed, so both are functions of \(t\): \(A(t)\) and \(r(t)\). Since the oil drop is forming a circle on the surface of the water, the area and radius are related by the traditional formula for the area of a circle: \(A=\pi\cdot r^2\).
The goal is to find how the rates of change with respect to time are related. Since \(A(t)\) and \(r(t)\) are functions of time, the use of implicit differentiation must be used to find \(\displaystyle{\frac{dA}{dt}}\) and \(\displaystyle{\frac{dr}{dt}}\). Since the area is constantly changing with respect to time, one specific instance is needed to analyze. For example, say when the radius is \(3\) cm, how fast is the area increasing? Or it could be how fast is the radius of the oil drop changing as the area reaches \(42\) square centimeters. After implicit differentiation, plug in any values for the variables. Use of how the variables are related might be needed to find values for all variables involved.
Tow small planes approach an airport, one flying due west at \(120\)mph and the other flying due north at \(150\)mph. Assuming they fly at the same constant elevation, how fast is the distance between the planes changing when the westbound plane is \(180\) miles from the airport and the northbound plane is \(225\) miles from the airport?
Let \(x\) denote the distance of the west bound plane to the airport. Let \(y\) denote the distance of the north bound plane to the airport. Let \(z\) denote the distance between the two planes.
| How are the variables related? | These three variables are sides of a right triangle, so by the Pythagorean Theorem: \(x^2+y^2=z^2\). |
| Use implicit differentiation to find the rates with respect to time: | \(2x\cdot\frac{dx}{dt}+2y\cdot\frac{dy}{dt}=2z\cdot\frac{dz}{dt}\) |
| Plug in all known values (may need to find values from original equation) | \(\begin{array}{l} x=180,\;y=225 \\ \implies z^2=(180)^2+(225)^2\implies z=288.14 \\ \frac{dx}{dt}=-120,\;\frac{dy}{dt}=-150 \\ \implies2(180)(-120)+2(225)(-150)=2(288.14)\cdot\frac{dz}{dt} \end{array}\) |
| Solve for appropriate rate: | \(\begin{array}{l} \frac{dz}{dt}=\frac{2(180)(-120)+2(225)(-150)}{2(288.14)} \\ \frac{dz}{dt}=-192.09\;\color{green}{\checkmark} \end{array}\) |
| The distance between the planes is decreasing at \(192.09\)mph. | |
NOTE: In the above problem, the rates of each plane are negative since the distance from the plane to the airport is decreasing.