Solving Related Rates Problems

Introduction

Key Concepts

Understanding Related Rates

Related rates problems deal with determining the rate at which one quantity changes in relation to another when both are functions of time. These problems are ubiquitous in various fields such as physics, engineering, and economics, where multiple variables are interdependent.

Prerequisites for Solving Related Rates Problems

Before tackling related rates problems, students must have a solid understanding of the following:

• Differentiation: Proficiency in taking derivatives of functions with respect to a variable, typically time ($t$).
• Implicit Differentiation: Ability to differentiate equations involving multiple variables without explicitly solving for one variable in terms of another.
• Chain Rule: Application of the chain rule to handle composite functions where one variable depends on another, which in turn depends on time.

Steps to Solve Related Rates Problems

Solving related rates problems generally involves the following systematic approach:

1. Identify the Variables: Determine which quantities are changing with respect to time and assign variables to them.
2. Express the Relationship: Formulate an equation that relates the variables using geometric or physical principles.
3. Differentiation: Differentiate both sides of the equation with respect to time ($t$) using appropriate differentiation rules.
4. Solve for the Desired Rate: Substitute known values into the differentiated equation and solve for the unknown rate.

Example Problem

Problem Statement: A ladder 10 feet long is leaning against a wall. If the bottom of the ladder slides away from the wall at a rate of 1.5 feet per second, how fast is the top of the ladder descending when the bottom is 6 feet from the wall?

Solution:

1. Identify the Variables:
   • Let $x$ be the distance of the bottom of the ladder from the wall.
   • Let $y$ be the height of the top of the ladder above the ground.
   • Given: $\frac{dx}{dt} = 1.5$ ft/s.
   • Find: $\frac{dy}{dt}$ when $x = 6$ ft.
2. Express the Relationship:

   The ladder forms a right triangle with the wall and the ground. Applying the Pythagorean theorem:

   $$x^2 + y^2 = 10^2$$
3. Differentiation:

   Differentiate both sides with respect to time ($t$):

   $$2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0$$
4. Solve for the Desired Rate:

   Solve for $\frac{dy}{dt}$:

   $$\frac{dy}{dt} = -\frac{x \frac{dx}{dt}}{y}$$

   First, find $y$ when $x = 6$ ft:

   $$6^2 + y^2 = 100 \Rightarrow y^2 = 64 \Rightarrow y = 8 \text{ ft}$$

   Substitute the known values:

   $$\frac{dy}{dt} = -\frac{6 \times 1.5}{8} = -\frac{9}{8} = -1.125 \text{ ft/s}$$

   The negative sign indicates that the top of the ladder is descending at 1.125 feet per second when the bottom is 6 feet from the wall.

Applications of Related Rates

Related rates have a wide array of applications in various disciplines:

• Physics: Calculating the speed of an object in motion, such as a falling balloon or a car accelerating down a highway.
• Engineering: Designing systems where multiple components interact dynamically, like in fluid dynamics or structural analysis.
• Economics: Modeling how different economic indicators change in relation to each other over time.
• Biology: Understanding rates of population growth or the spread of diseases.

Common Pitfalls and How to Avoid Them

Students often encounter challenges when dealing with related rates problems. Here are some common mistakes and tips to avoid them:

• Mistake: Not clearly identifying all the variables involved.
• Tip: Carefully read the problem and label all quantities with corresponding variables before forming equations.
• Mistake: Forgetting to apply the chain rule when differentiating implicitly.
• Tip: Practice implicit differentiation and ensure that each variable dependent on time is differentiated accordingly.
• Mistake: Misinterpreting the sign of the rate (positive vs. negative).
• Tip: Analyze the physical meaning of the rate to determine its direction (increasing or decreasing).

Advanced Techniques

As students become more comfortable with related rates problems, they can explore advanced techniques to tackle more complex scenarios:

• Multiple Related Variables: Handling problems where more than two variables are interdependent.
• Higher-Order Rates: Considering rates of change of rates of change, such as acceleration.
• Optimization Problems: Combining related rates with optimization to find maximum or minimum values under certain constraints.

Additional Example

Problem Statement: A spherical balloon is being inflated so that its volume increases at a rate of $100\pi$ cubic inches per minute. How fast is the radius of the balloon increasing when the radius is 5 inches?

Solution:

1. Identify the Variables:
   • Let $V$ be the volume of the balloon.
   • Let $r$ be the radius of the balloon.
   • Given: $\frac{dV}{dt} = 100\pi$ in³/min.
   • Find: $\frac{dr}{dt}$ when $r = 5$ in.
2. Express the Relationship:

   The volume of a sphere is given by:

   $$V = \frac{4}{3}\pi r^3$$
3. Differentiation:

   Differentiate both sides with respect to time ($t$):

   $$\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$$
4. Solve for the Desired Rate:

   Solve for $\frac{dr}{dt}$:

   $$\frac{dr}{dt} = \frac{\frac{dV}{dt}}{4\pi r^2}$$

   Substitute the known values:

   $$\frac{dr}{dt} = \frac{100\pi}{4\pi (5)^2} = \frac{100\pi}{100\pi} = 1 \text{ in/min}$$

   The radius of the balloon is increasing at a rate of 1 inch per minute when the radius is 5 inches.

Leveraging Technology

While understanding the mathematical foundations is crucial, leveraging technology can enhance the learning experience:

• Graphing Calculators: Useful for visualizing functions and their rates of change over time.
• Software Tools: Applications like Desmos or GeoGebra can help model related rates scenarios dynamically.
• Online Resources: Platforms offering interactive tutorials and practice problems can provide additional support outside the classroom.

Practice Problems

Engaging with a variety of practice problems is essential for mastering related rates. Here are some problems to test your understanding:

• Problem 1: A conical tank is being filled with water at a rate of $50$ cubic feet per minute. If the height of the tank is 10 feet and the radius is 5 feet, at what rate is the water level rising when the water is 4 feet deep?
• Problem 2: A shadow is being cast by a man walking away from a streetlight. The streetlight is 15 feet tall, and the man is 6 feet tall. If the man walks away from the streetlight at 2 feet per second, how fast is the tip of his shadow moving?
• Problem 3: A hot air balloon rises at a rate of 4 feet per second. At the same time, a point on the ground moves horizontally away from the balloon at 3 feet per second. How fast is the distance between the balloon and the point increasing after 5 seconds?

Comparison Table

Summary and Key Takeaways