### Solving Related Rates Problems: AP Calculus AB/BC Study Guide

#### Welcome, Calculus Conquerors!

Grab your calculators and your thinking caps because we’re diving deep into the world of related rates! Yes, it's time to apply your calculus superpowers to real-world problems. Prepare to tackle situations where rates of change are tangled together like your earphones after a jog. Let’s untangle this mess, shall we? 🎧✨

Related rates problems are like a big school dance where one variable changes its rate, and suddenly all the other variables have to follow suit. These problems usually involve figuring out how the rate of one quantity's change affects the rate of another's. Imagine a waltz where time is the rhythm that keeps everyone in step. Often, these scenarios include geometric shapes that morph over time, like an expanding balloon or a melting ice sculpture.

Follow these steps to master any related rates problem like a calculus Jedi:

**1. Read the Problem Carefully:** First things first, don't just skim through the problem like it’s the terms and conditions on a software update. Identify key values and relationships. Feel free to underline, circle, or even doodle next to them if it helps. 📜🔍

**2. Draw a Diagram:** Whip out your inner artist and sketch the scenario. Visualizing the problem often clarifies how variables interrelate. Plus, who doesn’t like drawing in math class? 🎨✏️

**3. Set up an Equation:** Use the given information to write an equation that ties the variables together. This usually involves some solid geometric knowledge or formulas. Think of it like a recipe where all ingredients must meet in one harmonious dish. 🍲

**4. Implicit Differentiation:** Differentiate the equation with respect to time (t). This is where calculus flexes its muscles. Remember to use the chain rule, treating each variable as a function of time. 💪🌀

**5. Substitute Known Values:** Plug in the values and rates of change you know into the differentiated equation. It’s like substituting ingredients in your cookie recipe—except this won't (probably) burn down your kitchen. 🍪➡️🍰

**6. Solve for the Desired Rate:** Solve the resulting equation to find the rate you’re interested in. Make sure your final units make sense (because minutes per elephant would just be confusing). 🚀🔍

#### Let’s Tackle Some Example Problems!

**1. Expanding Rectangle**

Imagine a rectangular garden happily expanding in the spring sun. One side is 6 feet (short side), and the other is 8 feet (long side). The short side is growing at 2 feet per minute. How fast is the garden's area increasing?

Crucial Info:

- Length (l) = 8 ft
- Width (w) = 6 ft
- Change in width (( \frac{dw}{dt} )) = 2 ft/min

Formula:

- Area ( A ) = length (\times) width ( (l \times w) )

Since the length isn't changing:

- ( \frac{dl}{dt} = 0 )

Differentiate Area with Respect to Time (t):

- ( A' = 8 \times \frac{d}{dt}(w) \times \frac{dw}{dt} )
- ( A' = 8 \times 1 \times 2 )
- ( A' = 16 \text{ ft}^2/\text{min} )

So, the area of the garden is growing at 16 square feet per minute. 🌱⏱️

**2. Sliding Ladder**

Picture a 13-meter ladder (we’ll call it Larry) leaning against a wall. The bottom of Larry is sliding away at 3 m/s. How fast is the top of Larry sliding down when the bottom is 5 meters from the wall?

Vital Data:

- ( x ) (distance from wall to bottom of ladder) = 5 m
- ( \frac{dx}{dt} ) = 3 m/s
- Ladder length ( (z) ) = 13 m constant

Key Equation:

- By the Pythagorean theorem: ( x^2 + y^2 = z^2 ) where ( y ) = height from the ground to the top of the ladder.

Differentiate Implicitly with Respect to Time (t):

- ( 2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0 )

Substitute Known Values and Solve for ( \frac{dy}{dt} ):

- Use ( x = 5 ) and solve for ( y ) using ( y = \sqrt{13^2 - 5^2} = 12 )
- Plug in: ( 2(5)(3) + 2(12) \frac{dy}{dt} = 0 )
- Simplify: ( 30 + 24 \frac{dy}{dt} = 0 )
- ( \frac{dy}{dt} = \frac{-30}{24} = - \frac{5}{4} \text{ m/s} )

Therefore, Larry's top is sliding down at a rate of ( - \frac{5}{4} \text{ m/s} ) or 1.25 meters per second when the bottom is 5 meters from the wall. 📏🏃

#### Key Terms You Can't Miss

**Conical Water Tank**: Imagine an upside-down ice cream cone for storing water. 🍦**Depth**: How deep you’ll have to dive to find Nemo. 🐠**Height**: How high you’ll get if you climb Jack’s beanstalk. 🪜**Proportion**: Like twins dressing identically, ensuring balance and equality. 👯♂️**Radius**: Half of your favorite pizza's distance from center to crust. 🍕**Rate of Change**: How fast you say "Yikes!" when your favorite show gets canceled. 📺**Related Rates**: How changing one thing messes with another, like dominos falling. 🃏**Volume Formula**: For when you need to know how much ice cream fits in a cone. 🍦**Volume of a Cone**: ( V = \frac{1}{3} \pi r^2 h ). Part of every sane dessert lover’s survival kit.

#### Conclusion

Great job, Calc Champions! 🙌 You now hold the magical key to solving related rates problems! These step-by-step methods are sure to help you navigate the twists and turns of the AP Calculus exam like a pro. Keep practicing, and soon, related rates will be as easy as pi (pun absolutely intended). 🥧🎉

Keep calm and differentiate on! 🚀🎓