The ratio of time taken by ice on the surface of ponds or lakes to become triple the thickness is

Options

A. 1 : 2
B. 1 : 3
C. 1 : 1
D. 1 : 9

Answer

To solve the problem of finding the ratio of time taken by ice on the surface of ponds or lakes to become triple its thickness, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Heat Flow**:
The rate of heat flow through a material can be expressed using Fourier's law of heat conduction:
\[ q = -k \frac{\Delta T}{\Delta x} \]
where \( q \) is the rate of heat flow, \( k \) is the thermal conductivity, \( \Delta T \) is the temperature difference across the material, and \( \Delta x \) is the thickness of the material.

2. **Relate Time to Thickness**:
The time taken for the ice to reach a certain thickness can be derived from the heat flow equation. If we denote the time taken to reach a thickness \( x \) as \( t_1 \) and the time taken to reach a thickness \( 3x \) as \( t_2 \), we can express these times in terms of the thickness:
\[ t_1 \propto x \quad \text{and} \quad t_2 \propto 3x \]

3. **Set Up the Ratio**:
Since time is directly proportional to thickness, we can write:
\[ t_1 = k \cdot x \quad \text{and} \quad t_2 = k \cdot 3x \]
where \( k \) is a constant that depends on the material properties and conditions.

4. **Calculate the Ratio**:
Now, we can find the ratio of \( t_1 \) to \( t_2 \):
\[ \frac{t_1}{t_2} = \frac{k \cdot x}{k \cdot 3x} = \frac{1}{3} \]

5. **Final Result**:
Therefore, the ratio of the time taken by the ice on the surface to become triple its thickness is:
\[ \text{Ratio} = 1 : 3 \]