b
(b)Let specific gravities of concrete and saw dust are \({\rho _1}\) and \({\rho _2}\) respectively.
According to principle of floatation weight of whole sphere = upthrust on the sphere
\(\frac{4}{3}\pi ({R^3} - {r^3}){\rho _1}g + \frac{4}{3}\pi {r^3}{\rho _2}g = \frac{4}{3}\pi {R^3} \times 1 \times g\)
==> \({R^3}{\rho _1} - {r^3}{\rho _1} + {r^3}{\rho _2} = {R^3}\)
==> \({R^3}({\rho _1} - 1) = {r^3}({\rho _1} - {\rho _2})\) ==> \(\frac{{{R^3}}}{{{r^3}}} = \frac{{{\rho _1} - {\rho _2}}}{{{\rho _1} - 1}}\)
==> \(\frac{{{R^3} - {r^3}}}{{{r^3}}} = \frac{{{\rho _1} - {\rho _2} - {\rho _1} + 1}}{{{\rho _1} - 1}}\)
==> \(\frac{{({R^3} - {r^3}){\rho _1}}}{{{r^3}{\rho _2}}} = \left( {\frac{{1 - {\rho _2}}}{{{\rho _1} - 1}}} \right)\;\frac{{{\rho _1}}}{{{\rho _2}}}\)
==> \(\frac{{{\rm{Mass of concrete }}}}{{{\rm{Mass of saw dust}}}} = \left( {\frac{{1 - 0.3}}{{2.4 - 1}}} \right) \times \frac{{2.4}}{{0.3}} = 4\)