AP Calc AB: Related Rates Problems in 5 Steps (with 2 worked examples)

Related rates problems are where many AP Calc AB students lose 4–6 FRQ points. Not because the calculus is hard, but because they skip the setup step that makes everything else mechanical. Here’s the 5-step framework Tae uses with his AP Calc students.

The 5 steps, every time

  1. Identify the variables. What's changing? What's constant?
  2. Find the equation that relates them. Geometry, distance, area, volume, Pythagoras, similar triangles.
  3. Differentiate with respect to time (implicitly). Every variable's time derivative gets a dt notation.
  4. Plug in the values. The question gives you specific values at the moment of interest. Plug them in after differentiating, not before.
  5. Solve for the unknown rate. Report with units.

Example 1 — The ladder problem

A 13-ft ladder leans against a wall. The base slides away from the wall at 2 ft/s. How fast is the top sliding down the wall when the base is 5 ft from the wall?

Step 1: variables are x (base distance from wall, increasing) and y (top height, decreasing). Both change with time.

Step 2: Pythagoras: x² + y² = 169.

Step 3: differentiate with respect to t: 2x(dx/dt) + 2y(dy/dt) = 0, so dy/dt = −(x/y)(dx/dt).

Step 4: at x = 5, y = √(169 − 25) = 12. dx/dt = 2. So dy/dt = −(5/12)(2) = −5/6 ft/s.

Step 5: the top is sliding down at 5/6 ft/s. Negative sign confirms direction (downward).

Example 2 — The cone problem

A conical tank with radius 5 ft and height 10 ft is being filled with water at 3 ft³/min. How fast is the water level rising when the depth is 4 ft?

Step 1: variables are h (water depth) and V (water volume), both increasing. Radius of water surface r also changes.

Step 2: The water forms a cone similar to the tank. By similar triangles, r/h = 5/10, so r = h/2. Volume of cone: V = (1/3)πr²h = (1/3)π(h/2)²h = (π/12)h³.

Step 3: dV/dt = (π/4)h²(dh/dt).

Step 4: at h = 4, dV/dt = 3. So 3 = (π/4)(16)(dh/dt), giving dh/dt = 3/(4π) ≈ 0.239 ft/min.

Step 5: the water level rises at 3/(4π) ft/min when the depth is 4 ft.

Related deep-dive walkthroughs

Step-by-step guides Tae uses with students to lock in the highest-leverage AP problem types:

Common mistakes

  1. Plugging in values before differentiating. If you write “V = (π/4)(16) = 4π,” you've lost the variable. Differentiate first, plug in second.
  2. Forgetting the similar-triangles substitution. In the cone problem, you must eliminate r in terms of h before differentiating.
  3. Wrong sign on the rate. If a quantity is decreasing, its derivative is negative. Use that in the equations, not after.
  4. No units in the answer. AP graders deduct points if you write “5/6” instead of “5/6 ft/s”.

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