4.5 Solving Related Rates Problems

• The first step in solving a related rates problem is to identify the variables and their relationships
• Steps to identify variables and their relationships in related rates problems

  1️⃣ List all variables involved

  2️⃣ Determine the relationships between variables

  3️⃣ Identify the unknown variable

  4️⃣ Express the rate of change of the unknown variable
• The relationships between variables in related rates problems are always linear.

  False
• Steps to differentiate the relationship equation with respect to time

  1️⃣ Take the derivative of both sides

  2️⃣ Apply the chain rule if necessary

  3️⃣ Use the product rule if necessary
• In a related rates problem, the first step is to identify the variables
• Steps to differentiate a relationship with respect to time in related rates problems:

  1️⃣ Take the derivative of both sides of the relationship equation

  2️⃣ Apply the chain rule if variables depend on time

  3️⃣ Use the product rule if necessary
• After differentiating, substitute the known values and solve for the unknown rate
• Match the solution checks with their descriptions:

  Ensure correct units ↔️ Verify the units of the calculated rate of change

  Confirm the sign ↔️ Check if the rate of change is increasing or decreasing

  Check the magnitude ↔️ Ensure the numerical value is reasonable
• Steps in analyzing the solution of a related rates problem

  1️⃣ Interpret what the rate of change tells us about the system's behavior

  2️⃣ Relate the rate of change to other variables and their rates of change

  3️⃣ Consider the practical significance of the rate of change in a real-world context
• When checking a solution, it's important to ensure the units of the rate of change are correct
• Why is it important to check the magnitude of the rate of change in a related rates problem?

  To ensure reasonableness
• The rate of change in a related rates problem is independent of other variables.

  False
• The rate of change in a related rates problem always indicates an increase in the system's quantity.

  False
• A positive rate of change for the volume of a cylinder indicates that the volume is increasing.

  True
• A positive rate of change indicates that the volume of a cylinder is increasing
• The rate of change in a related rates problem is independent of the rates of change of other variables.

  False
• What is the second step in solving a related rates problem?

  Differentiate with respect to time
• When solving related rates problems, it is necessary to list only the known variables.

  False
• What is the goal of identifying the unknown variable in a related rates problem?

  Find its rate of change
• The volume of a cylinder is given by the formula $V =$$\pi r^{2} h$, where r is the radius and h is the height
• What is the purpose of differentiating the relationship equation with respect to time in a related rates problem?

  Relate the rates of change
• The derivative of $V =$$\pi r^{2} h$ with respect to time is $\frac{dV}{dt} =$ \pi \left(2r\frac{dr}{dt}h + r^{2}\frac{dh}{dt}\right).


  True
• To find the rates of change in a related rates problem, we differentiate the relationship equation with respect to time.


  True
• What is the derivative of $V =$$\pi r^{2} h$ with respect to $t$?


  $\frac{dV}{dt} =$ \pi \left(2r\frac{dr}{dt}h + r^{2}\frac{dh}{dt}\right)
• After solving for the unknown rate, it is important to check and analyze the solution for correctness.


  True
• Steps in checking the solution of a related rates problem

  1️⃣ Ensure the units of the rate of change are correct

  2️⃣ Confirm the rate of change has the expected sign

  3️⃣ Check that the magnitude of the rate of change is reasonable
• The rate of change in a related rates problem must always be positive.

  False
• The calculated rate of change tells us about the behavior of the system
• What should you consider when analyzing the practical significance of a rate of change?

  Real-world context
• What is one example of the practical significance of a rate of change in a real-world context?

  Monitoring container growth
• What does the rate of change tell us about the behavior of a system in related rates problems?

  System dynamics
• Match the concept with its description in the analysis of related rates solutions:

  Practical significance ↔️ Real-world application of the rate of change

  System behavior ↔️ Changes in the system over time