Question

Three cubes of metal whose edges are in the ratio $3:4:5$ are melted and converted into a single cube whose diagonal is $12\sqrt{3}cm$. Find the edges of the three cubes.

Answer 1

the edge of three cube $3x,4x$ & $5x$ respectively.

volume of $3x$ cube $=(3x{)}^3=27x^3$

$4x$ cube $=(4x{)}^3=64x^3$

$5x$ cube $=(5x{)}^3=125x^3$

total volume of three cubes $=27x^3+64x^3+125x^3$$=216x^3$

volume of new cube = volume of three cubes

$\therefore a^3=216x^3$

$\therefore a=6x$

$\to$ diagonal of new cube $=12\sqrt{3}$

$\therefore \sqrt{a^2+a^2+a^2}=12\sqrt{3}$

$\therefore \sqrt{3a^2}=12\sqrt{3}$

$\therefore \sqrt{3}a=12\sqrt{3}$

$\therefore a=\frac{12\sqrt{3}}{\sqrt{3}}$

$\therefore a=12$

$\to a=6x$

$\therefore 12=6x$

$x=2$

The edge of three cubes $3\times 2:4\times 2:5\times 2$

$6:8:10$

$3x=3\times 2=6cm$

$4x=4\times 2=8cm$

$5x=5\times 2=10cm$