1. **Problem:** Find the approximate change in volume $V$ of a cube of side $n$ meters caused by increasing the side by 1%. 2. **Formula and explanation:** The volume of a cube is given by $$V = n^3.$$ When the side length changes by a small amount $\Delta n$, the approximate change in volume $\Delta V$ can be found using the differential $$dV = \frac{dV}{dn} dn = 3n^2 dn.$$ 3. **Calculate the change in side length:** Increasing the side by 1% means $$dn = 0.01 n.$$ 4. **Calculate the approximate change in volume:** Substitute $dn$ into the differential formula: $$dV = 3n^2 \times 0.01 n = 0.03 n^3.$$ 5. **Interpretation:** The volume increases approximately by 3% when the side length increases by 1%. **Final answer:** The approximate change in volume is $$\boxed{0.03 n^3}.$$