Q.An edge of a variable cube is increasing at the rate of $3 \text{ cm/s}$. How fast is the volume of the cube increasing when the edge is $10 \text{ cm}$ long?

Concept: Related Rates — we differentiate the volume with respect to time using the chain rule.

Let the edge length be $x$ cm and the volume be $V = x^3$ cm³.

Given $\frac{dx}{dt} = 3$ cm/s. We need $\frac{dV}{dt}$ when $x = 10$ cm.

Differentiate $V$ with respect to $t$:

$\frac{dV}{dt} = 3x^2 \cdot \frac{dx}{dt}$ …

This is a classic related rates problem — a staple in calculus. The idea is simple: when one quantity (the edge length $s$) changes with time, another quantity that depends on it (the volume $V$) also changes. We connect their rates of change using differentiation with respect to time $t$.

The key step is always: write the relationship between the quantities, then differentiate both sides with respect to time. Do not plug in the given numbers until after you differentiate — that’s a common trap.

1. Write the relationship. For a cube of edge length $s$, the volume is

2. Differentiate with respect to time $t$. Since both $V$ and $s$ are functions of $t$, we use the chain rule:

This equation tells us: the rate at which volume grows depends on the current edge length and the rate at which the edge itself grows.

3. Identify the given rates and the instant.

   We are told:

   • $\frac{ds}{dt} = 3$ cm/s (constant rate of increase of the edge).
   • We want $\frac{dV}{dt}$ when $s = 10$ cm.

4. Substitute the values.

   $\frac{dV}{dt} = 3 \cdot (10)^2 \cdot 3 = 3 \cdot 100 \cdot 3 = 900.$ …

Method: Related Rates for a Single Geometric Relation (Volume)

The same core technique as area-vs-radius problems, applied to a volume that depends on a single linear dimension.

Steps

Step 1: Write the volume formula in terms of the changing dimension

For a cube of edge $x$, $V=x^3$.

Step 2: Differentiate both sides with respect to time

The chain rule is what brings in the factor of $\frac{dx}{dt}$ — $x$ itself is a function of time. …

Common Mistakes

Mistake 1: Substituting the edge length before differentiating

Writing $V=10^3=1000$ first and then differentiating a constant gives zero for the rate — the differentiation of $V=x^3$ must happen while $x$ is still symbolic, with numbers substituted only in the final step.

Mistake 2: Missing the factor of 3 from the power rule …