Related Rates

Related rates problems involve finding the rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. Typically, these quantities and their rates are changing with respect to time. The core idea is using the Chain Rule to connect these changing rates.

Related rates are used to model situations where multiple variables are changing over time, and you want to find the rate of change of one variable given the rates of others. Common examples include:

• The rate at which the volume of a balloon is changing given the rate its radius changes.
• The rate at which the water level in a tank is changing as water is poured in.
• The speed at which the top of a ladder slides down a wall as the base slides away from the wall.
• The rate at which the distance between two moving objects changes.

Solving related rates problems typically involves these steps:

1. **Understand the Problem:** Read carefully and visualize the situation.
2. **Identify Variables:** Assign symbols to all quantities that are changing and the quantity whose rate you need to find.
3. **Draw a Diagram:** Sketch the situation and label the variables and given values (at a specific instant).
4. **Write the Given & Unknown Rates:** List the rates of change you know and the rate you need to find (using Leibniz notation like dV/dt, dr/dt, etc.).
5. **Find an Equation:** Write a mathematical equation that relates the variables from step 2 (that is true at *any* time, not just the specific instant). This equation will usually come from geometry or other fundamental principles.
6. **Differentiate Implicitly:** Differentiate both sides of the equation from step 5 with respect to time (t). This is where the Chain Rule is crucial.
7. **Substitute:** Plug in the known values for variables and rates at the specific instant given in the problem.
8. **Solve:** Solve the resulting equation for the unknown rate.

The Chain Rule is fundamental because variables are related to each other (e.g., volume and radius of a sphere), but their rates of change are usually given with respect to a third variable, time (t). If you have a relationship between two variables, say V and r, and you want to differentiate it with respect to t, the Chain Rule states dV/dt = dV/dr * dr/dt. This allows you to connect the rate of change of V with respect to t to the rate of change of r with respect to t.

Common mistakes include:

• Substituting the specific values *before* differentiating the main equation (the equation must be true for all time, not just the instant).
• Forgetting to use the Chain Rule when differentiating variables with respect to time (e.g., treating d/dt(r²) as 2r instead of 2r * dr/dt).
• Using incorrect geometric formulas or relationships.
• Getting units wrong or inconsistent.
• Algebraic errors when solving for the unknown rate.

Drawing a diagram and carefully listing knowns and unknowns helps avoid these issues.