Related rates problems

Definition

Related rates problems involve finding the rate at which one quantity changes in relation to another quantity that is also changing. These problems are typically framed in a real-world context and require the use of implicit differentiation to relate the different rates of change, often utilizing the chain rule from calculus.

5 Must Know Facts For Your Next Test

1. To solve related rates problems, first identify all the variables involved and how they are related through an equation.
2. It's crucial to differentiate the relationship between the variables with respect to time to find their rates of change.
3. Clearly define each variable and what it represents before applying derivatives, as this ensures clarity in your solution.
4. Units matter! Always keep track of units when solving related rates problems to ensure that your answers make sense.
5. Once you have found the relationship and differentiated, substitute known values into your derived equation to solve for the unknown rate.

Review Questions

• How do you approach a related rates problem, and what are the key steps you should follow?
  • To approach a related rates problem, start by identifying all the variables involved and writing down any relationships between them. Next, express these relationships mathematically with an equation. After that, differentiate both sides of the equation with respect to time using implicit differentiation. Finally, substitute known values into your differentiated equation to solve for the unknown rate of change you're interested in.
• Discuss how implicit differentiation is applied in related rates problems and provide an example scenario.
  • Implicit differentiation is applied in related rates problems when the relationship between variables is given in a form where one variable is not explicitly solved for another. For example, consider a scenario where a balloon's radius is increasing at a certain rate. The volume of the balloon can be expressed as V = (4/3)πr³. By differentiating both sides with respect to time using implicit differentiation, we can relate the rate of change of volume to the rate of change of radius, allowing us to find how quickly the volume is changing as the radius increases.
• Evaluate a specific related rates problem: If a ladder 10 feet long is leaning against a wall and the bottom is being pulled away from the wall at 2 feet per second, how fast is the top of the ladder descending when the base is 6 feet away from the wall?
  • To solve this problem, we start with the Pythagorean theorem: x² + y² = 10², where x is the distance from the wall and y is the height of the ladder on the wall. Differentiating with respect to time gives us 2x(dx/dt) + 2y(dy/dt) = 0. Plugging in x = 6 feet and dx/dt = 2 feet/second allows us to solve for dy/dt, which represents how fast the top of the ladder is descending. After calculating, we find that dy/dt = -2.4 feet/second, indicating that the top of the ladder is descending at that rate when it is 6 feet away from the wall.

Related terms

Implicit Differentiation: A technique used to differentiate equations that define y implicitly as a function of x, allowing for the calculation of derivatives when y is not isolated.

Chain Rule: A fundamental rule in calculus for finding the derivative of a composite function, essential for solving related rates problems since many quantities depend on one another.

Differential Equation: An equation that relates a function with its derivatives, often used to model real-world phenomena involving rates of change.