Question

A 5 cm thick ice block is there on the surface of the water in a lake. The temperature of the air is $-{{10}^{\circ }}C$; how much time will it take to double the thickness of the block? $(L=80cal/g,K=0.004Erg/s,d=0.92gc{{m}^{-3}})$
A. 1 hour
B. 191 hour
C. 19.1 hour
D. 1.91 hour

Answer 1

Hint:In this question, we have to find the time taken to double the thickness of the block. For this we have to use the formula of thickness of block and by putting the values and simplifying it, we are able to know the time taken.

Formula Used:
Thickness of the block is determined by,
$t=\dfrac{\rho L}{2K\theta }({{x}_{2}}^{2}-{{x}_{1}}^{2})$
Where $\rho $ is the density of ice, L is the latent heat of fusion, K is the conductivity of ice and $\theta $ is the temperature of air.

Complete step by step solution:
We have given 5 cm thick block
So the thickness of ice block = 5 cm
And the temperature of the air = $-{{10}^{\circ }}C$
We have to find the time to double the thickness of the block.
We know thickness of the block is determined by,
$t=\dfrac{\rho L}{2K\theta }({{x}_{2}}^{2}-{{x}_{1}}^{2})$………………………………… (1)

Now we put all the values in equation (1), we get
$t=\dfrac{0.92\times 80}{2\times 0.004\times 10}({{(10)}^{2}}-{{(5)}^{2}}) \\ $
Simplifying the above equation, we get
$t=6900\,s \\ $
$\Rightarrow t=\dfrac{69000}{3600}hr \\ $
$\therefore t=19.1\,hr$
Thus it will take 19.1 hr to double the thickness of the block.

Thus, option C is the correct answer.

Note: In these types of questions, students make mistakes in remembering the formula to be used to solve the question. As these types of questions have quite complex calculations, students must take care while doing calculations.