Question

There is the formation of the layer of snow $x\,cm $ thick on water, when the temperature of the air is $ - {\theta ^ \circ }\,C $ (less than a freezing point). The thickness of the layer increases from $x $ to $y $ in the time $t $, then the value of $t $ is given by:
(A) $\dfrac{{\left( {x + y} \right)\left( {x - y} \right)\rho L}}{{2k\theta }} $
(B) $\dfrac{{\left( {x - y} \right)\rho L}}{{2k\theta }} $
(C) $\dfrac{{\left( {x + y} \right)\left( {x - y} \right)\rho L}}{{k\theta }} $
(D) $\dfrac{{\left( {x - y} \right)\rho Lk}}{{2\theta }} $

Answer 1

Hint The value of the time can be determined by the formula of the time equation of the thermal property of the matter of the radiation, then the thickness of the layer is substituted as $x $ to $y $ and then the value of the time is determined.
Useful formula
The time equation of the thermal property of the matter of the radiation is given as,
 $t = \dfrac{{\rho L}}{{2k\theta }}\left( {{x_2}^2 - {x_1}^2} \right) $
Where, $t $ is the time taken, $\rho $ is the density of the material, $L $ is the length of the layer, $k $ is the constant, $\theta $ is the temperature, ${x_2} $ to ${x_1} $ is the thickness of the layer.

Complete step by step answer
Given that,
The formation of the layer of snow $x\,cm $ thick on water,
The temperature of the air is $ - {\theta ^ \circ }\,C $,
The thickness of the layer increases from $x $ to $y $.
Now,
The time equation of the thermal property of the matter of the radiation is given as,
 $t = \dfrac{{\rho L}}{{2k\theta }}\left( {{x_2}^2 - {x_1}^2} \right) $
By substituting the thickness values in the above equation, then the above equation is written as,
 $t = \dfrac{{\rho L}}{{2k\theta }}\left( {{x^2} - {y^2}} \right) $
By using the mathematical formula of $\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right) $, then the above equation is written as,
 $t = \dfrac{{\rho L}}{{2k\theta }}\left( {x + y} \right)\left( {x - y} \right) $
By rearranging the terms in the above equation, then the above equation is written as,
 $t = \dfrac{{\left( {x + y} \right)\left( {x - y} \right)\rho L}}{{2k\theta }} $
Thus, the above equation shows the value of the time.

Hence, the option (A) is the correct answer.

Note The time taken is directly proportional to the density and the length and inversely proportional to the temperature. As the density and the length increases, then the time taken also increases. As the density and the length decreases, then the time taken also decreases.