Explanation

(a) Let $V$ be the volume of solid ; $d$ be its density and $m$ be its mass ; if g coefficient of volume expansion of liquid, then
Density at temperature $t_{1}\ is,\ d_{1} = \frac{d_{0}}{1 + \gamma t_{1}}$
Density at temperature $t_{2}\ is,\ d_{2}$ = $\frac{d_{0}}{1 + \gamma t_{2}}$
According to Archimede’s principle,
$f_{1}V\ d_{1} = m = f_{2}\ Vd_{2}$
Or $\frac{d_{1}}{d_{2}}\ = \frac{f_{2}}{f_{1}} = \ \frac{d_{0}}{\left( 1 + \gamma t_{1} \right)}\frac{\left( 1 + \gamma t_{2} \right)}{d_{0}}$
Or $f_{1} + f_{1}\gamma t_{2} = f_{2} + f_{2}\ \gamma t_{1}$
$f_{1} - f_{2}\ = \ \gamma\left( f_{2}t_{1} - f_{1}t_{2} \right)$
$\gamma\ = \ \frac{(f_{1} - f_{2})}{f_{2}t_{1} - f_{1}t_{2}}$