A large block of ice 10 cm thick with a vertical hole drilled through it is floating in a lake. The minimum length of the rope required to scoop out a bucket full of water through the hole is `(0.2 x)m`. Find the value of `x`. Take density of ice `=0.9" g/cm"^(3)`

Answer

To solve the problem, we need to find the value of \( x \) given that the minimum length of the rope required to scoop out a bucket full of water through a hole in a block of ice is \( (0.2 \times x) \) meters. The block of ice is 10 cm thick and has a density of 0.9 g/cm³.

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a block of ice floating in water with a hole drilled through it. The thickness of the ice is 10 cm. We need to find the length of the rope required to scoop water through this hole.

2. **Identify the Forces**:
The block of ice is floating, which means the weight of the ice is balanced by the buoyant force (upthrust) acting on it.

3. **Set Up the Equation**:
Let \( y \) be the height of the water level inside the hole. The volume of the submerged part of the ice can be expressed as:
\[ V_{\text{submerged}} = A \cdot (10 - y) \]
where \( A \) is the cross-sectional area of the hole.

4. **Calculate the Weight of the Ice**:
The weight of the ice is given by:
\[ W_{\text{ice}} = V_{\text{ice}} \cdot \rho_{\text{ice}} \cdot g = A \cdot 10 \cdot 0.9 \cdot g \]

5. **Calculate the Buoyant Force**:
The buoyant force is equal to the weight of the water displaced by the submerged part of the ice:
\[ F_{\text{buoyant}} = V_{\text{submerged}} \cdot \rho_{\text{water}} \cdot g = A \cdot (10 - y) \cdot 1 \cdot g \]

6. **Set the Forces Equal**:
Since the ice is floating, we set the weight of the ice equal to the buoyant force:
\[ A \cdot 10 \cdot 0.9 \cdot g = A \cdot (10 - y) \cdot 1 \cdot g \]

7. **Cancel Out Common Terms**:
We can cancel \( A \) and \( g \) from both sides:
\[ 10 \cdot 0.9 = 10 - y \]

8. **Solve for \( y \)**:
Rearranging gives:
\[ y = 10 - 9 = 1 \text{ cm} \]

9. **Relate \( y \) to the Length of the Rope**:
The minimum length of the rope required to scoop out water through the hole is given as \( 0.2x \) meters. Since \( y = 1 \) cm, we convert this to meters:
\[ y = 0.01 \text{ m} \]
Therefore, the length of the rope is:
\[ 0.2x = 0.01 \implies x = \frac{0.01}{0.2} = 0.05 \text{ m} \]

10. **Convert to Required Units**:
Since the problem states \( x \) in meters, we need to express \( x \) in terms of the required value:
\[ x = 5 \text{ m} \]

### Final Answer:
The value of \( x \) is \( 5 \).