If the areas of three adjacent faces of a rectangular block are in the ratio of 2 : 3 : 4 and its volume is 9000 cu. cm; then the length of the shortest side is

### Concept & Finding Dimensions from Face Areas Using the standard identity $V = \sqrt{xyz}$ where $x, y, z$ are the areas of adjacent faces, we can find the actual areas given their ratio. Once the face areas are known, individual sides can be found since $l = V / (bh)$, $b = V / (lh)$, and $h = V / (lb)$. ### Step-by-Step Solution 1. **Given:** Ratio of areas $x : y : z = 2 : 3 : 4$. Volume $V = 9000$. Let the areas be $2k, 3k, 4k$. 2. **Find the Constant $k$:** $$ V = \sqrt{x \times y \times z} $$ $$ 9000 = \sqrt{2k \times 3k \times 4k} $$ $$ 9000 = \sqrt{24k^3} $$ Square both sides: $$ 81000000 = 24k^3 $$ $$ k^3 = \frac{81000000}{24} = 3375000 $$ $$ k = \sqrt[3]{3375 \times 1000} = 15 \times 10 = 150 $$ 3. **Calculate Face Areas:** The areas are $x = 300$, $y = 450$, $z = 600$. 4. **Find the Shortest Side:** The side lengths are $\frac{V}{\text{Area}}$. The shortest side corresponds to the largest face area. $$ \text{Shortest side} = \frac{V}{\text{Largest Area}} = \frac{9000}{600} = 15 \text{ cm} $$ ### Exam Strategy & Shortcut To find the shortest side directly, divide the volume by the largest face area. You don't need to calculate all three sides. ### Common Pitfall Calculating $k$ but forgetting to apply it back to find the largest area, or mistakenly dividing the volume by the smallest area to find the shortest side (inverse relationship). ### Final Answer Therefore, the correct answer is **15 cm**.