Ice starts forming in lake with water at `0^(@)C` and when the atmospheric temperature is `-10^(@)C`. If the time taken for `1 cm` of ice be `7` hours. Find the time taken for the thickness of ice to change from `1 cm` to `2 cm`

Options

A. 7 hours
B. 14 hours
C. 10.5 hours
D. 21 hours

Answer

To solve the problem of finding the time taken for the thickness of ice to change from 1 cm to 2 cm, we can follow these steps:

### Step 1: Understand the relationship between thickness and time
The thickness of ice formed is related to the time taken for its formation. The relationship can be expressed as:
\[ t \propto x^2 \]
where \( t \) is the time taken and \( x \) is the thickness of the ice.

### Step 2: Establish the constant of proportionality
From the relationship above, we can express the time taken to form a thickness \( x \) as:
\[ t = \frac{x^2}{2c} \]
where \( c \) is a constant that depends on the material properties and conditions.

### Step 3: Calculate the time for 1 cm thickness
Given that the time taken to form 1 cm (0.01 m) of ice is 7 hours, we can substitute \( x = 1 \) cm into the equation:
\[ t_1 = \frac{1^2}{2c} = 7 \text{ hours} \]
This implies:
\[ \frac{1}{2c} = 7 \implies 2c = \frac{1}{7} \implies c = \frac{1}{14} \]

### Step 4: Calculate the time for 2 cm thickness
Now, we can calculate the time taken to form 2 cm (0.02 m) of ice:
\[ t_2 = \frac{2^2}{2c} = \frac{4}{2c} \]
Substituting \( c = \frac{1}{14} \):
\[ t_2 = \frac{4}{2 \times \frac{1}{14}} = \frac{4 \times 14}{2} = 28 \text{ hours} \]

### Step 5: Find the time difference
To find the time taken for the thickness of ice to change from 1 cm to 2 cm, we calculate:
\[ \Delta t = t_2 - t_1 = 28 \text{ hours} - 7 \text{ hours} = 21 \text{ hours} \]

### Final Answer
The time taken for the thickness of ice to change from 1 cm to 2 cm is **21 hours**.

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