A solid whose volume does not change with temperature floats in a liquid. For two different temperatures t₁ and t₂ of the liquid, fractions f₁ and f₂ of the volume of the solid remain submerged in the liquid. The coefficient of volume expansion of the liquid is equal to

The correct option is:
f₁-f₂/f₂t₁-f₁t₂

Step-by-step explanation:

Let $V$ be the total volume of the solid. Since the volume of the solid does not change with temperature, $V$ remains constant.
Let ${\rho }_1$ and ${\rho }_2$ be the densities of the liquid at temperatures $t_1$ and $t_2$, respectively.

For a floating body, the principle of floatation states that the weight of the floating body is equal to the buoyant force (the weight of the displaced liquid).
Weight of the solid, $W=V_{\mathrm{submerged}}\times {\rho }_{\mathrm{liquid}}\times g$

At temperature $t_1$, the fraction of the volume submerged is $f_1$, so the submerged volume is $f_{1}V$. Therefore:
$W=f_{1}V{\rho }_{1}g$

At temperature $t_2$, the fraction of the volume submerged is $f_2$, so the submerged volume is $f_{2}V$. Therefore:
$W=f_{2}V{\rho }_{2}g$

Equating the two expressions for the weight $W$:
$f_{1}V{\rho }_{1}g=f_{2}V{\rho }_{2}g$
Simplifying by dividing both sides by $Vg$:
$f_1{\rho }_1=f_2{\rho }_2$

The density of a liquid varies with temperature according to the relation:
$\rho =\frac{{\rho }_0}{1+\gamma t}$
where ${\rho }_0$ is the density of the liquid at $0$ degrees, and $\gamma$ is the coefficient of volume expansion of the liquid.

Substituting this relation for ${\rho }_1$ and ${\rho }_2$:
$f_1\left(\frac{{\rho }_0}{1+\gamma t_1}\right)=f_2\left(\frac{{\rho }_0}{1+\gamma t_2}\right)$

Dividing both sides by ${\rho }_0$:
$\frac{f_1}{1+\gamma t_1}=\frac{f_2}{1+\gamma t_2}$

Cross-multiplying to solve for $\gamma$:
$f_1(1+\gamma t_2)=f_2(1+\gamma t_1)$
$f_1+f_1\gamma t_2=f_2+f_2\gamma t_1$

Rearranging the terms containing $\gamma$ to one side:
$f_1\gamma t_2-f_2\gamma t_1=f_2-f_1$
Factoring out $\gamma$:
$\gamma (f_1{t}_2-f_2{t}_1)=f_2-f_1$

Solving for $\gamma$:
$\gamma =\frac{f_2-f_1}{f_1{t}_2-f_2{t}_1}$

Multiply both the numerator and denominator by -1:
$\gamma =\frac{f_1-f_2}{f_2{t}_1-f_1{t}_2}$