Ice starts forming on the surface of lake and takes 8 hours to form a layer of 1 cm thick. How much time will it take to increase the thickness of layer to 2 cm ?

Options

A. 8 hours
B. Less than 8 hours
C. Between 8 to 16 hours
D. More than 16 hours

Explanation

To solve the problem of how long it will take to increase the thickness of the ice layer from 1 cm to 2 cm, we can follow these steps:

### Step 1: Understand the relationship between time and thickness
The time taken to freeze the ice is directly proportional to the square of the thickness of the ice. This means that if we denote the thickness of the ice as \( d \) and the time taken as \( t \), we can express this relationship as:
\[ t \propto d^2 \]

### Step 2: Establish the known values
From the problem, we know that:
- It takes 8 hours to form a 1 cm thick layer of ice.
- Therefore, we can write:
\[ t_1 = 8 \text{ hours} \quad \text{and} \quad d_1 = 1 \text{ cm} \]

### Step 3: Calculate the time for 2 cm thickness
We want to find the time \( t_2 \) for a thickness \( d_2 = 2 \text{ cm} \). Using the proportionality established in Step 1, we can set up the following equation:
\[ \frac{t_2}{t_1} = \frac{d_2^2}{d_1^2} \]

Substituting the known values:
\[ \frac{t_2}{8} = \frac{(2)^2}{(1)^2} \]

### Step 4: Simplify the equation
Calculating the right side:
\[ \frac{t_2}{8} = \frac{4}{1} \]
\[ t_2 = 8 \times 4 \]

### Step 5: Calculate the final time
Now, we can calculate:
\[ t_2 = 32 \text{ hours} \]

Thus, it will take **32 hours** to increase the thickness of the ice layer to 2 cm.

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