Q.A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm³ /min. When the thickness of ice is 5 cm, find the rate at which the thickness of ice decreases.

Related Rates: When Two Things Change Together

Imagine you're blowing up a balloon. Your lungs push air in at a certain rate — say, 3 cubic centimetres per second. As the balloon grows, its radius increases. The question is: how fast is the radius increasing at the exact moment when the radius is 5 cm?

That's a related rates problem. Two quantities — the volume $V$ and the radius $r$ — are linked by a geometric formula ($V = \frac{4}{3}\pi r^3$). You know how fast one is changing ($dV/dt = 3$), and you want to find how fast the other is changing ($dr/dt$) at a specific instant.

The core idea is simple: if two quantities are connected by an equation, their rates of change are also connected — by the derivative of that equation.

The Precise Statement

Let $x$ and $y$ be two quantities that both depend on time $t$, and suppose they satisfy some equation $F(x, y) = 0$ (or $y = f(x)$, etc.). Then:

1. Differentiate both sides of the equation with respect to time $t$.
2. Use the chain rule wherever you see a variable that depends on $t$.
3. The result is an equation linking $dx/dt$ and $dy/dt$.

That's it. The "related rates" are $dx/dt$ and $dy/dt$, and the chain rule is the tool that connects them.

The Chain Rule in Action

In the balloon example, the equation is $V = \frac{4}{3}\pi r^3$. Differentiate both sides with respect to $t$:

The right side is a function of $r$, and $r$ itself depends on $t$. By the chain rule:

Now plug in what you know: $dV/dt = 3$ and $r = 5$:

So:

That's the answer. The radius is growing at about 0.0095 cm/s when the balloon's radius is 5 cm.

The General Recipe

For any related rates problem, follow these steps:

1. Identify all changing quantities and assign them variables. Note which rates you know and which you need.
2. Write an equation that relates the quantities at any time $t$ (not just at the instant of interest).
3. Differentiate implicitly with respect to $t$. Use the chain rule for every variable that depends on $t$.
4. Substitute the known values (including the specific instant's measurements) into the rate equation.
5. Solve for the unknown rate.

A Second Example: The Ladder Problem …