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Unit 4.5 – Solving Related Rates Problems

AP® Calculus AB & BC | Every Step for Full Exam Credit

Related rates are among the most common FRQs on the AP® exam! You’ll connect the rates at which different quantities change with time—using diagrams, equations, and calculus. Problems almost always start as story contexts.

📋 Classic 6-Step Related Rates Process

  1. Draw a diagram and label all changing quantities (use variables, with units!).
  2. Define GIVEN and FIND (what rates and values are known, and what rate is sought).
  3. Write an equation that relates all the variables involved in the problem.
  4. Differentiate with respect to time (\(t\)): use the chain rule & implicit differentiation on both sides of the equation.
  5. Plug in the known values (quantities & rates) for the specific instant in question.
  6. Solve for the required rate. Write your answer with units and in context.

🔢 Must-Know Related Rates Formulas & Patterns

  • Circle Area: \( A = \pi r^2 \) → \( \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \)
  • Sphere Volume: \( V = \frac{4}{3} \pi r^3\) → \( \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt} \)
  • Pythagorean Theorem: \( x^2 + y^2 = h^2 \) → \( 2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 2h\frac{dh}{dt} \)
  • Cylinder Volume: \( V = \pi r^2 h \) → \( \frac{dV}{dt} = 2\pi r h \frac{dr}{dt} + \pi r^2 \frac{dh}{dt} \)
  • Rectangle Area: \( A = lw \) → \( \frac{dA}{dt} = l\frac{dw}{dt} + w\frac{dl}{dt} \)

💡 Expert Short Notes & Tricks

  • Only substitute numbers after differentiating—unless a variable represents a constant quantity!
  • Always write units for each rate: e.g., \(\frac{dx}{dt}\) in m/s, \(\frac{dA}{dt}\) in cm²/min.
  • If a variable is not changing (e.g., a cylinder with a fixed radius), then its rate of change is zero (e.g., \(\frac{dr}{dt}=0\)).
  • Draw a clear, labeled diagram up front—this helps organize the problem and is often expected for full credit.
  • Direction matters: use positive values for increasing quantities/rates and negative values for decreasing ones.

📖 AP®-Style Worked Example

Expanding Circular Ripple (Full Setup)
Question: The radius \(r\) of a circular ripple expands at a rate of 2 cm/sec. How fast is the area increasing when the radius is 5 cm?
  1. Given: The rate of change of the radius is \(\frac{dr}{dt}=2\) cm/sec. We want to find the rate of change of the area at the instant when \(r=5\) cm.
  2. Find: \(\frac{dA}{dt}\).
  3. Equation: The formula for the area of a circle is \(A = \pi r^2\).
  4. Differentiate: Differentiating with respect to time gives \(\frac{dA}{dt} = 2\pi r \frac{dr}{dt}\).
  5. Substitute: Plug in the known values: \(\frac{dA}{dt} = 2\pi (5)(2) = 20\pi\).
  6. Statement: "When the radius is 5 cm, the area is increasing at a rate of \(20\pi\) cm²/sec."

📝 Practice Problems

Try These Yourself:
  • A 10 ft ladder is leaning against a wall. The top of the ladder is sliding down the wall at 3 ft/sec. How fast is the base of the ladder moving away from the wall when the top is 6 ft above the ground?
  • Sand is poured to make a conical pile. The height of the pile is always twice the radius of its base. If the height is increasing at 0.4 m/min, how fast is the volume of sand changing when the height is 5 m?
  • A sphere’s radius is shrinking at a rate of 0.1 cm/min. Find the rate at which the volume is decreasing when the radius is 7 cm.
Setups/Answers:
  • Use \(x^2 + y^2 = 10^2\). Given \(y=6\) ft and \(\frac{dy}{dt}=-3\) ft/sec. First, find \(x\) (\(x=8\) ft). Then differentiate to get \(2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0\) and solve for \(\frac{dx}{dt}\).
  • Use \(V=\frac{1}{3}\pi r^2 h\). Since \(h=2r\), you can write \(V\) in terms of just \(h\): \(V=\frac{1}{12}\pi h^3\). Differentiate to get \(\frac{dV}{dt} = \frac{1}{4}\pi h^2 \frac{dh}{dt}\). Plug in \(h=5\) and \(\frac{dh}{dt}=0.4\).
  • Use \(V = \frac{4}{3}\pi r^3\). Differentiate to get \(\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}\). Plug in \(r=7\) and \(\frac{dr}{dt}=-0.1\).

✏️ AP® Exam Success Tips for Related Rates

  • Draw and label your diagram, and state all variables and their units clearly.
  • Do not plug in values for changing quantities until after you have differentiated!
  • State the final answer as a complete sentence with correct units.
  • Clearly indicate the direction of change (e.g., "increasing," "decreasing") based on the sign of the rate.
  • Box your final answer and explicitly show your differentiation step for full credit on FRQs.