Related Rates

What Are Related Rates?

In a related rates problem, multiple quantities (like lengths, areas, volumes, and angles) are all changing with respect to time $t$. The relationship between the quantities gives you an equation, and differentiating that equation with respect to $t$ gives you a relationship between their rates of change.

For example, if a circle's radius $r$ is growing over time, the area $A = \pi r^2$ is also growing. Differentiating with respect to $t$:

$\frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt}$

This relates the rate of change of area ($\frac{dA}{dt}$) to the rate of change of radius ($\frac{dr}{dt}$).

The General Strategy

Every related rates problem follows the same five-step process:

Step 1: Draw a diagram and label variables. Assign variable names to all changing quantities. Use constants for quantities that don't change.

Step 2: Write down what you know and what you want. Identify the given rates (e.g., $\frac{dr}{dt} = 3$ cm/s) and the unknown rate you need to find.

Step 3: Write an equation relating the variables. This is the critical step. The equation comes from geometry, trigonometry, or physics. Common relationships include:

• Pythagorean theorem: $a^2 + b^2 = c^2$
• Area/volume formulas: $A = \pi r^2$, $V = \frac{4}{3}\pi r^3$, $V = \frac{1}{3}\pi r^2 h$
• Similar triangles
• Trigonometric relationships

Step 4: Differentiate both sides with respect to $t$. Use implicit differentiation. Every variable is a function of $t$, so every variable gets a $\frac{d}{dt}$ (chain rule).

Step 5: Substitute known values and solve. Plug in the known rates and the specific values of the variables at the instant in question. Solve for the unknown rate.

Important: Do NOT Substitute Before Differentiating

This is the most critical rule. If a variable is changing with time, do NOT plug in its specific value before differentiating. Substituting a number for a variable makes it a constant, and its derivative becomes zero. Only substitute after you have differentiated.

For example, if the problem says "at the moment when $r = 5$," keep $r$ as a variable in your equation, differentiate, and only then plug in $r = 5$.

Implicit Differentiation Review

Since every variable depends on $t$, differentiating an equation like $x^2 + y^2 = 100$ gives:

$2x \cdot \frac{dx}{dt} + 2y \cdot \frac{dy}{dt} = 0$

Each term picks up a $\frac{d(\text{variable})}{dt}$ factor — this is the chain rule applied to each variable as a function of $t$.

Common Geometric Relationships

Circle: $A = \pi r^2$, $C = 2\pi r$

Sphere: $V = \frac{4}{3}\pi r^3$, $S = 4\pi r^2$

Cone: $V = \frac{1}{3}\pi r^2 h$ (often need to eliminate $r$ or $h$ using similar triangles)

Cylinder: $V = \pi r^2 h$

Pythagorean theorem: $a^2 + b^2 = c^2$ (ladders, distances)

Similar triangles: $\frac{a}{b} = \frac{c}{d}$ (shadows, cones)

Right-triangle trigonometry: $\tan \theta = \frac{\text{opp}}{\text{adj}}$

Eliminating Variables with Similar Triangles

In cone problems, the volume formula $V = \frac{1}{3}\pi r^2 h$ has two variables ($r$ and $h$). If the cone has a fixed shape (like a conical tank), similar triangles give a relationship between $r$ and $h$:

$\frac{r}{h} = \frac{R}{H}$

where $R$ and $H$ are the cone's full radius and height. This lets you write $r = \frac{R}{H} h$ and substitute, reducing to one variable.

Tip 1: Draw a Clear Diagram Every Time

Even if you think you understand the problem, a diagram prevents errors. Label all variables, mark what's constant, and indicate which quantities are changing.

Tip 2: Assign Units and Check Dimensional Consistency

If the radius is in cm and the rate is in cm/s, then $\frac{dA}{dt}$ should be in cm²/s and $\frac{dV}{dt}$ in cm³/s. Checking units catches many errors.

Tip 3: Write the Equation Before Reading Rates

Find the geometric/algebraic relationship between variables first. Then identify which rates you know and which you need. This prevents you from trying to force rates into the wrong equation.

Tip 4: Differentiate Carefully — Don't Rush

Implicit differentiation with respect to $t$ is where most errors occur. Differentiate each term systematically. Remember: constants differentiate to zero; variables get $\frac{d}{dt}$ factors.

Tip 5: State the Answer with Units and Sign Interpretation

On free-response, explicitly state your answer with units. A negative rate means the quantity is decreasing. If the problem asks "how fast is the distance decreasing," and you get $\frac{dD}{dt} = -3$, state: "The distance is decreasing at 3 ft/s."