The temperature of the water of pond is `0^@ C ` while that of the surrounding atmosphere is `-20^@ C`. If the density of ice is p, coefficient of thermal conductivity is k and latent heat of melting is L then the thickness Z of ice layer formed increases as function of time t as

Options

A. `Z^2 = (60 k)/(rho L)t`
B. `Z=sqrt((40 k)/(rho L))l`
C. `Z^2 =( 40 K)/(rho L) sqrt(t)`
D. `z^2 =(400 k)/( rho L) t`

Explanation

To find the thickness \( Z \) of the ice layer formed over time \( t \), we can follow these steps:

### Step 1: Understand the Heat Transfer
The heat current \( \frac{dq}{dt} \) through the ice layer can be expressed using Fourier's law of heat conduction:
\[ \frac{dq}{dt} = kA \frac{\Delta T}{x} \]
where:
- \( k \) is the coefficient of thermal conductivity,
- \( A \) is the area,
- \( \Delta T \) is the temperature difference (which is \( 0 - (-20) = 20 \) °C),
- \( x \) is the thickness of the ice layer.

### Step 2: Set Up the Heat Loss Equation
The heat loss \( dq \) when ice of thickness \( dx \) is formed can be expressed as:
\[ dq = mL = \rho A dx L \]
where:
- \( \rho \) is the density of ice,
- \( L \) is the latent heat of melting,
- \( m \) is the mass of the ice formed.

### Step 3: Equate the Two Expressions
From the two expressions for \( dq \):
\[ \rho A dx L = kA \frac{20}{x} dt \]
We can cancel \( A \) from both sides:
\[ \rho dx L = k \frac{20}{x} dt \]

### Step 4: Rearrange and Separate Variables
Rearranging gives:
\[ \rho L dx = k \frac{20}{x} dt \]
Now, we can separate the variables:
\[ \frac{x \, dx}{\rho L} = \frac{20 k}{dt} \]

### Step 5: Integrate Both Sides
Integrate both sides. The left side integrates from \( 0 \) to \( z \) and the right side from \( 0 \) to \( t \):
\[ \int_0^z x \, dx = \int_0^t \frac{20 k}{\rho L} dt \]
This gives:
\[ \frac{z^2}{2} = \frac{20 k}{\rho L} t \]

### Step 6: Solve for \( z \)
Multiplying both sides by 2:
\[ z^2 = \frac{40 k}{\rho L} t \]
Taking the square root:
\[ z = \sqrt{\frac{40 k}{\rho L} t} \]

### Final Expression
Thus, the thickness \( Z \) of the ice layer formed as a function of time \( t \) is:
\[ Z = \sqrt{\frac{40 k}{\rho L} t} \]