Related Rates

A ladder slides down a wall. Air is pumped into a balloon. Water pours into a cone-shaped tank. In each of these, several quantities change at once, and an equation keeps them linked. Related rates problems use that link: if you know how fast one quantity is changing, the derivative tells you how fast another one is changing. They’re one of the most common applications of the derivative, and a regular on AP Calculus exams.

What Are Related Rates?

Suppose a circle’s radius is growing over time. Its area is growing too. Both are functions of time , so differentiate both sides with respect to , using the chain rule on :

That’s a related rates equation. It links the two rates and , so knowing one gives you the other.

The key step is the same one as in implicit differentiation, except the variable you differentiate with respect to is time. Every changing quantity is secretly a function of , so each one picks up its own rate:

How to Solve Related Rates Problems: 5 Steps

1. Draw a picture and give every changing quantity a variable name.
2. List the rates. Write the rate you’re given and the rate you want, both as derivatives with respect to . A quantity that’s decreasing has a negative rate.
3. Write an equation that relates the quantities. Geometry usually supplies it: the Pythagorean theorem, an area or volume formula, or similar triangles.
4. Differentiate both sides with respect to .
5. Substitute and solve. Only now plug in the values at the moment the question asks about, then solve for the unknown rate. Include units.

Step 5 is where most mistakes happen. Plugging in numbers before differentiating turns the variables into constants, and their rates disappear.

Example 1: A Spreading Ripple

A stone dropped in a pond makes a circular ripple whose radius grows at 2 cm/s. How fast is the area growing when the radius is 10 cm?

Steps 1 and 2 — quantities and rates: radius , area . Given ; find when .

Step 3 — the equation:

Step 4 — differentiate with respect to :

Step 5 — substitute:

The radius grows at a steady rate, but the area grows faster and faster, because each new ring has a longer circumference.

Example 2: Blowing Up a Balloon

This is a version of the “bubble gum problem”, where a growing bubble is treated as a sphere. Air is pumped into a spherical balloon at 100 cm³/s. How fast is the radius increasing when the radius is 5 cm?

Steps 1 and 2: ; find when .

Step 3 — the volume of a sphere:

Step 4 — differentiate:

Step 5 — substitute and solve:

At , the same calculation gives cm/s. The air flows in at the same rate, but a bigger balloon needs much more air to grow the same amount.

Example 3: The Sliding Ladder

A 10 ft ladder leans against a wall. The bottom slides away from the wall at 2 ft/s. How fast is the top sliding down when the bottom is 6 ft from the wall?

Steps 1 and 2: let be the distance from the wall to the bottom, and the height of the top. Given ; find when .

Step 3 — Pythagorean theorem:

Step 4 — differentiate:

Step 5 — find , then substitute. When , :

The negative sign means is decreasing: the top slides down at 1.5 ft/s. The bottom moves at a steady 2 ft/s, but the top doesn’t. When , and ft/s. The top speeds up as it nears the ground.

Example 4: Filling a Conical Tank

Water pours at 2 m³/min into a tank shaped like an upside-down cone, 4 m tall with a top radius of 2 m. How fast is the water level rising when the water is 3 m deep?

Steps 1 and 2: water depth , surface radius , volume . Given ; find when .

Step 3 — volume of a cone, then remove . has two variables, but you only know a rate for one of them. The water forms a smaller cone similar to the tank, so , which gives :

Step 4 — differentiate:

Step 5 — substitute and solve:

Eliminating a variable before differentiating is a common move. Use a relationship that is true at every moment, like the similar triangles here, not a value that is only true at one instant.

Example 5: A Shadow Problem

A 6 ft tall person walks away from a 15 ft lamppost at 5 ft/s. How fast is the person’s shadow getting longer?

Steps 1 and 2: is the person’s distance from the post and the shadow length. Given ; find .

Step 3 — similar triangles. The big triangle (lamppost) and the small one (person) share the angle at the tip of the shadow:

Step 4 — differentiate:

Step 5 — substitute:

The answer doesn’t depend on where the person is, so the question doesn’t need to give a distance. A follow-up question often asks how fast the tip of the shadow moves. The tip is at distance from the post, so it moves at ft/s.

Example 6: Two Cars and an Intersection

Car A is 3 miles north of an intersection, driving south toward it at 30 mph. Car B is 4 miles east of the intersection, driving east at 40 mph. Is the distance between the cars increasing or decreasing, and how fast?

Steps 1 and 2: is car A’s distance north, is car B’s distance east, is the distance between them. Car A’s distance is shrinking, so , and . Find when , .

Step 3 — Pythagorean theorem:

Step 4 — differentiate and cancel the 2s:

Step 5 — substitute. At this moment :

The distance is increasing at 14 mph. Car A is closing in, but car B is pulling away faster.

Common Mistakes

• Substituting before differentiating. In the ladder problem, writing first makes a constant, so vanishes. Keep every changing quantity as a variable until after step 4.
• Forgetting the rate factor. , not . Every variable that changes gets its own .
• Wrong signs. A quantity that is shrinking has a negative rate. Car A’s distance to the intersection decreases, so its rate is , not .
• Treating a fixed quantity as a variable, or the other way round. The ladder’s length (10) and the tank’s dimensions (2 and 4) are constants. The water depth and the shadow are not.
• Missing values at the instant. You often need to find a quantity the problem didn’t give, like in the ladder problem, from the original equation.

Practice Problems

Try each one before opening the answer. The practice set has 12 more with instant checking.

Problem 1. The side of a square grows at 3 cm/s. How fast is the area growing when the side is 5 cm?

Problem 2. A circle’s area grows at cm²/s. How fast is the radius growing when the radius is 3 cm?

Problem 3. The edge of a cube grows at 0.5 cm/s. How fast is the volume growing when the edge is 4 cm?

Problem 4. A 13 ft ladder leans against a wall, and its bottom slides away at 3 ft/s. How fast is the top sliding down when the bottom is 5 ft from the wall?

Problem 5. A snowball melts so that its radius shrinks at 0.2 cm/min. How fast is its volume changing when the radius is 10 cm?

Problem 6. A plane flies horizontally at an altitude of 6 km and a speed of 800 km/h, passing directly over a radar station. How fast is its distance from the station increasing when that distance is 10 km?

Related rates use the derivative to compare how fast things change. Next, the derivative goes the other way: using the tangent line to estimate values of a function near a point you know.