If the areas of the three adjacent faces of a cuboidal box are 120 cm$^2$, 72 cm$^2$ and 60 cm$^2$ respectively, then find the volume of the box.

### Concept & Volume from Adjacent Faces As derived from the geometric properties of a cuboid, if $x, y$, and $z$ are the areas of three adjacent faces, the volume $V$ is the square root of their product. $$ V = \sqrt{x \times y \times z} $$ ### Step-by-Step Solution 1. **Given:** $x = 120$, $y = 72$, $z = 60$. 2. **Apply Formula:** $$ V = \sqrt{120 \times 72 \times 60} $$ 3. **Simplify the Calculation:** Instead of multiplying everything out, factorize to find perfect squares. $$ 120 \times 60 = 7200 $$ $$ V = \sqrt{7200 \times 72} $$ $$ V = \sqrt{100 \times 72 \times 72} $$ $$ V = 10 \times 72 = 720 \text{ cm}^3 $$ ### Exam Strategy & Shortcut Never multiply large numbers into a massive single integer when taking a square root. Group them efficiently. Spotting $120 \times 60 = 72 \times 100$ immediately pairs with the other $72$ to make extraction trivial. ### Common Pitfall Getting lost in multiplying $120 \times 72 \times 60 = 518400$ and then struggling to calculate the square root of a large number without a calculator. ### Final Answer Therefore, the correct answer is **720 cm$^3$**.