Question

Three cubes whose edges are in the ratio 3:4:5 are melted and converted into a single cube whose diagonal is 12cm. find the edges of the three cubes

Answer 1

let the sides of cubes be 3x,4x and 5x
now all 3 are melted to form single one of edge a...
so there volumes will remain same
$(3x) {}^{3} + (4x) {}^{3} + (5x) {}^{3} = {a}^{3}$
now diagnol of cube=12
$a \sqrt{3} = 12 \\ {a}^{3} = \frac{4}{ \sqrt{ 3} }$
put aboxe
${x}^{3} (27 + 64 + 125) = \frac{4}{ \sqrt{3} }$
x=
$x = \frac{4}{ \sqrt{3}(216) }$
put above in sides

Answer 2

$\bigstar{\huge{\underline{\mathfrak{\red{Answer}}}}}$

Let the edges of three cubes (in cm) be 3x, 4x and 5x respectively. Let a be the side of the new cube so formed after melting.

Volume of new cube = Sum of the volumes of three smaller cubes

$\sf{{a}^{3} = {(3x)}^{3} + {(4x)}^{3} + {(5x)}^{3} = {216x}^{3} = {(6x)}^{3} }$

$\therefore$ $\sf{a = 6x}$

Diagonal of the new cube = 12√3cm

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

$\longrightarrow$$\sf{a \sqrt{3} = 12 \sqrt{3}}$

$\longrightarrow$$\sf{a = 12cm}$

$\longrightarrow$$\sf{6x = 12cm}$

$\therefore$ $\sf{x = 2cm}$

Hence, the ages of the three cubes are 6cm, 8cm and 10cm respectively.