Question

If the areas of the adjacent faces of a rectangular block are in the ratio 2 : 3 : 4 and its volume is 9000 $c{m}^3,$ then the length of the shortest edge is

• A. 30 cm
• B. 20 cm
• C. 15 cm
• D. 10 cm

Answer 1

The correct option is C

15 cm

$\text{Ratio in the areas of three adjacent faces of a cuboid}=2:3:4$

$\text{Volume}=9000c{m}^3$

$\text{Let the area of faces be 2x, 3x, 4x and}$

$\text{Let a, b, and c be the dimensions of the cuboid, then}$

$\therefore 2x=ab,3x=bc,4x=ca$

$\therefore ab\times bc\times ca=2x\times 3x\times 4x$

$a^2{b}^2{c}^2=24x^3$

$\text{But volume}=abc=9000c{m}^3$

$24x^3=(9000{)}^2$

$x^3=\frac{81000000}{24}=3375000$

$\therefore x=\sqrt[3]{33375000}=\sqrt[3]{3(150{)}^3}=150$

$\therefore ab=300$

$bc=450$

$ca=600$

$\because abc=9000$

$\text{Dividing one by one, we get}$

$c=30,a=20,b=15$

$\therefore \text{Shortest side}=15cm$