4.5 Solving Related Rates Problems

• One of the first steps in solving related rates problems is to identify the variables
• What is the purpose of differentiating the equation with respect to time in related rates problems?

  Find unknown rates
• Why is it important to check the solution and provide appropriate units in related rates problems?

  Ensure accuracy
• Steps to solve related rates problems

  1️⃣ Identify the given information

  2️⃣ Set up an equation

  3️⃣ Differentiate the equation
• If the radius of a circle is increasing at 2 cm/s, the rate of change of the area is $4\pi r$
• Expressing the relationship between variables in an equation is necessary for solving related rates problems.

  True
• What is the known rate in the example of a circle's radius increasing at 2 cm/s?

  $\frac{dr}{dt} =$$2$
• Identifying variables is the first step in solving related rates problems.


  True
• Match the type of equation with its description:

  Area ↔️ Relates dimensions to the area of geometric shapes

  Volume ↔️ Relates dimensions to the volume of geometric shapes

  Pythagorean Theorem ↔️ Relates the sides of a right triangle
• In the ladder example, what equation is used to relate the variables?

  Pythagorean Theorem
• What is the first step in differentiating an equation with respect to time in related rates problems?

  Apply the chain rule
• What is the result of differentiating $a^{2} +$$b^{2} =$$c^{2}$ with respect to time?


  $a \frac{da}{dt} +$$b \frac{db}{dt} =$$0$
• Steps to solve related rates problems

  1️⃣ Identify given information

  2️⃣ Set up an equation

  3️⃣ Differentiate the equation
• The equation relating the area of a circle to its radius is $A =$$\pi r^{2}$


  True
• To solve related rates problems, you must first identify the variables and their relationships.
• The equation for the volume of a cylinder is $V =$$\pi r^{2} h$


  True
• What is the formula for the area of a circle used in related rates problems?

  $A =$$\pi r^{2}$
• A ladder 13 feet long is leaning against a wall. If the base is sliding away at 2 ft/s, what is the equation relating the variables?

  $a^{2} +$$b^{2} =$$c^{2}$
• What do you substitute known values into in related rates problems after differentiating the equation?

  Differentiated equation
• In related rates problems, appropriate units must always be included in the final answer.


  True
• What do related rates problems involve understanding?

  Changing quantities over time
• Establishing a formula or equation is a critical step in solving related rates problems.

  True
• In related rates problems, after differentiating the equation, you must substitute given values
• Related rates problems involve understanding the relationship between two or more changing quantities.


  True
• What is the equation for the area of a circle?

  $A =$$\pi r^{2}$
• What is the first step in identifying variables and their relationships in related rates problems?

  List the variables
• If the volume of a cube is increasing, the variables are side length (x) and volume
• Steps to solve related rates problems

  1️⃣ Identify given information

  2️⃣ Set up an equation

  3️⃣ Differentiate the equation
• Why is establishing a formula or equation essential in related rates problems?

  Relate changing quantities
• The Pythagorean Theorem is used in related rates problems to relate the sides of a triangle
• Implicit differentiation ensures that each term includes a derivative with respect to time.


  True
• What is the first step in solving a related rates problem?

  Identify given information
• In a related rates example, the initial condition for the radius is 5 cm.
• What is the rate at which the area of a circle increases when the radius is 5 cm and increasing at 2 cm/s?

  $20\pi$ cm²/s
• Match the variable with its meaning in the context of a cylinder's volume:

  Volume ↔️ $V$

  Radius ↔️ $r$

  Height ↔️ $h$
• In related rates problems, establishing a formula or equation relates the variables to describe their relationship.
• The Pythagorean theorem is commonly used in related rates problems involving right triangles.


  True
• In related rates problems, the chain rule is applied to differentiate each variable term, e.g., \frac{d}{dt}(x^{n}) = nx^{n - 1} \frac{dx}{dt}</latex>.chain
• Differentiating the Pythagorean theorem with respect to time results in $a \frac{da}{dt} +$$b \frac{db}{dt} =$$0$.


  True
• If $\frac{db}{dt} =$$2$ and $a =$$12$, $b =$$5$, then $\frac{da}{dt} =$$- \frac{5}{6}$.5