4.5 Solving Related Rates Problems

    Cards (112)

    • What are related rates problems in calculus concerned with?
      Rate of change over time
    • Related rates problems often involve finding the rate of change of one variable given the rate of change of another
    • The equation for the area of a circle is A=A =πr2 \pi r^{2}.
    • What is the derivative of the area of a circle AA with respect to time tt?

      dAdt=\frac{dA}{dt} =2πrdrdt 2 \pi r \frac{dr}{dt}
    • If the radius rr of a circle is increasing at 2 cm/s, then \frac{dr}{dt} = 2
    • Related rates problems require identifying variables and their relationships.
    • Steps to identify variables and relationships in related rates problems
      1️⃣ Define each variable and its rate of change
      2️⃣ Write down the equation relating the variables
    • What is the equation relating the area AA and radius rr of a circle?

      A=A =πr2 \pi r^{2}
    • The equation A=A =πr2 \pi r^{2} relates the area AA of a circle to its radius
    • Drawing a diagram in related rates problems helps visualize the relationships between variables.
    • What is one key benefit of drawing a diagram in related rates problems?
      Clarifies the problem
    • Match the geometric property with the corresponding variable in a cone-shaped tank
      Height ↔️ h
      Radius ↔️ r
      Volume ↔️ V
    • Stating the given information in related rates problems ensures clarity and accuracy
    • What is an example of given information in a related rates problem?
      drdt=\frac{dr}{dt} =2 2
    • If drdt=\frac{dr}{dt} =2 2 cm/s, then \frac{dA}{dt} = 4 \pi r</latex> cm²/s.
    • Why is drawing a diagram useful in related rates problems?
      Visualizes the relationships
    • What is the equation relating the area AA and the radius rr of a circle?

      A=A =πr2 \pi r^{2}
    • The area of a circle is given by the equation A
    • Diagrams can help clarify the context of related rates problems.
    • Steps to use diagrams in related rates problems
      1️⃣ Visualize the problem
      2️⃣ Label variables and rates of change
      3️⃣ Clarify relationships between variables
    • What is the formula for the volume of a cone in terms of its radius rr and height hh?

      V=V =13πr2h \frac{1}{3} \pi r^{2} h
    • Stating the given information in related rates problems is essential for accuracy
    • If the radius of a circle is r=r =5 5 cm and its rate of change is drdt=\frac{dr}{dt} =2 2 cm/s, what information has been stated?

      Radius and rate of change
    • In related rates problems, stating what needs to be found provides a key objective for solving the problem.
    • Clearly stating what needs to be found in related rates problems is crucial for defining the objective
    • What is the area AA of a rectangle where the length ll is twice the width ww?

      A=A =l22 \frac{l^{2}}{2}
    • Establishing the equation relating the variables is a critical step in solving related rates problems.
    • To establish the equation in related rates problems, the first step is to identify known quantities
    • What are related rates problems in calculus designed to find?
      Rate of change of one variable
    • In related rates problems, if drdt=\frac{dr}{dt} =2 2 cm/s, then \frac{dA}{dt} = 4\pi r</latex> cm²/s. This result is obtained by differentiation
    • Steps to solve related rates problems
      1️⃣ Define variables and rates of change
      2️⃣ Write the equation relating variables
      3️⃣ Differentiate the equation with respect to time
      4️⃣ Substitute known values
      5️⃣ Solve for the target rate of change
    • Stating the given information in related rates problems ensures clarity and accuracy.
    • What are the values of rr and drdt\frac{dr}{dt} in the example of a circle with a changing radius?

      r=r =5 5 cm, drdt=\frac{dr}{dt} =2 2 cm/s
    • In related rates problems, the target rate of change is written using differential notation
    • The first step in stating given information is to identify the variables
    • What should numerical values in related rates problems always include?
      Units
    • Steps to state given information in related rates problems
      1️⃣ Identify variables
      2️⃣ Write down their values
      3️⃣ Use a table
    • Clearly stating what needs to be found is crucial for defining the objective of related rates problems.
    • The target rate of change is written using differential notation
    • What should the target rate of change always include?
      Units