Question

Three cubes of a metal whose edge are in ratio 3:4:5are melted and converted into a single cube whose diagonal is12√3cm.find the edge of three cubes

Answer 1

$diagonal \: of \: a \: cube = \sqrt{3} \times side \\ side = \frac{12 \sqrt{3} }{ \sqrt{3} } = 12cm \\ volume \: of \: cube \: = volume \: of \: all \: three \: cubes \: \\ ( {12}^{3}) = ( {3x})^{3} + ( {4x})^{3} + ( {5x})^{3} \\ 1728 = 27 {x}^{3} + 64 {x}^{3} + {125x}^{3} \\ 1728 = 216 {x}^{3} \\ x = \sqrt[3]{ \frac{1728}{216} } \\ x = \frac{12}{6} = 2 \\ \\ side \: of \: first \: cube \: = 3x = 3 \times 2 = 6cm \\ side \: of \: second \: cube = 4x = 4 \times 2 = 8cm \\ side \: of \: third \: cube = 5x = 5 \times 2 = 10cm$