Steps to solve related rates problems using the distance formula in calculus

📚 Introduction to Related Rates and the Distance Formula

Related rates problems involve finding the rate at which a quantity changes by relating it to other quantities whose rates of change are known. When the problem involves distances between moving objects, the distance formula becomes a crucial tool. This guide will walk you through the steps to solve these types of problems effectively.

📜 Historical Context

The concept of related rates evolved alongside calculus itself, primarily through the work of Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. Understanding how quantities change relative to one another is a cornerstone of calculus and is used extensively in physics, engineering, and economics.

🔑 Key Principles for Solving Related Rates Problems

• 🔍 Understand the Problem: Read the problem carefully and identify what rates are given and what rate you need to find. Draw a diagram if possible.
• ✏️ Identify Variables: Assign variables to all quantities that are functions of time ($t$).
• 📏 Establish a Relationship: Write an equation that relates the variables. In problems involving distances, this is often the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• ⏱️ Differentiate with Respect to Time: Use implicit differentiation to differentiate both sides of the equation with respect to time ($t$). Remember to apply the chain rule.
• ➕ Substitute Known Values: Plug in the given values for the variables and their rates of change.
• ✅ Solve for the Unknown Rate: Solve the resulting equation for the rate you're trying to find.
• 📢 State the Answer: Include the correct units in your final answer.

💡 Example 1: Two Cars Moving Away from an Intersection

Problem: Car A is traveling east at 30 mph and Car B is traveling north at 40 mph. Both are headed toward an intersection. At what rate is the distance between the two cars changing 2 hours after they pass through the intersection?

Solution:

1. Draw a Diagram: Draw a right triangle with the intersection at the right angle. Car A's distance east is $x$, Car B's distance north is $y$, and the distance between them is $d$.
2. Equation: $d = \sqrt{x^2 + y^2}$
3. Given: $\frac{dx}{dt} = 30$ mph, $\frac{dy}{dt} = 40$ mph. After 2 hours, $x = 30 \cdot 2 = 60$ miles, $y = 40 \cdot 2 = 80$ miles.
4. Differentiate: $\frac{dd}{dt} = \frac{1}{2\sqrt{x^2 + y^2}} (2x \frac{dx}{dt} + 2y \frac{dy}{dt}) = \frac{x \frac{dx}{dt} + y \frac{dy}{dt}}{\sqrt{x^2 + y^2}}$
5. Substitute: $\frac{dd}{dt} = \frac{60(30) + 80(40)}{\sqrt{60^2 + 80^2}} = \frac{1800 + 3200}{\sqrt{3600 + 6400}} = \frac{5000}{100} = 50$
6. Answer: The distance between the cars is increasing at 50 mph.

💡 Example 2: Airplane Flying Over a Radar Station

Problem: An airplane is flying horizontally at an altitude of 6 miles and a speed of 480 mph. It passes directly over a radar station. At what rate is the distance between the plane and the radar station increasing 30 minutes later?

Solution:

1. Draw a Diagram: Draw a right triangle with the radar station at one vertex. The altitude is constant at 6 miles. Let $x$ be the horizontal distance the plane has traveled from the station and $d$ be the distance between the plane and the radar station.
2. Equation: $d = \sqrt{x^2 + 6^2}$
3. Given: $\frac{dx}{dt} = 480$ mph. After 30 minutes (0.5 hours), $x = 480 \cdot 0.5 = 240$ miles.
4. Differentiate: $\frac{dd}{dt} = \frac{1}{2\sqrt{x^2 + 36}} (2x \frac{dx}{dt}) = \frac{x \frac{dx}{dt}}{\sqrt{x^2 + 36}}$
5. Substitute: $\frac{dd}{dt} = \frac{240(480)}{\sqrt{240^2 + 36}} = \frac{115200}{\sqrt{57600 + 36}} = \frac{115200}{\sqrt{57636}} \approx 479.85$
6. Answer: The distance between the plane and the radar station is increasing at approximately 479.85 mph.

📝 Practice Quiz

Try these problems to test your understanding:

1. 🚗 Two cars start from the same point. One travels south at 60 mph and the other travels west at 25 mph. How fast is the distance between them increasing after 2 hours?
2. 🚀 A rocket is launched vertically and tracked by a radar station located 3 miles from the launch site. When the rocket is 4 miles above the ground, its speed is 2000 mph. What is the rate of change of the distance between the rocket and the radar station at that instant?

🎓 Conclusion

Solving related rates problems using the distance formula requires a systematic approach. By carefully identifying variables, establishing relationships, differentiating with respect to time, and substituting known values, you can find the rate of change you seek. Practice is key to mastering these types of problems, so work through various examples to build your confidence.