A baby's bath should be at a temperature of `30^(circ) C`. There is `10 kg` water at `12^(circ) C` in the bath tub. How much water (in kg ) with temperature `50^(circ) C` should be added to achieve the desired temperature?

Answer

To solve the problem of how much water at 50°C should be added to 10 kg of water at 12°C to achieve a final temperature of 30°C, we can use the principle of conservation of energy, which states that the heat lost by the hot water will be equal to the heat gained by the cold water.

### Step-by-Step Solution:

1. **Identify the known values**:
- Mass of cold water (m1) = 10 kg
- Initial temperature of cold water (T1) = 12°C
- Final temperature (Tf) = 30°C
- Temperature of hot water (T2) = 50°C
- Mass of hot water (m2) = ? (this is what we need to find)

2. **Write the heat gained by cold water**:
The heat gained by the cold water can be expressed as:
\[ Q_{cold} = m_1 \cdot C_w \cdot (T_f - T_1) \]
Where \(C_w\) is the specific heat capacity of water (which will cancel out later).

3. **Write the heat lost by hot water**:
The heat lost by the hot water can be expressed as:
\[ Q_{hot} = m_2 \cdot C_w \cdot (T_2 - T_f) \]

4. **Set the heat gained equal to the heat lost**:
According to the principle of conservation of energy:
\[ Q_{cold} = Q_{hot} \]
Substituting the expressions we derived:
\[ m_1 \cdot C_w \cdot (T_f - T_1) = m_2 \cdot C_w \cdot (T_2 - T_f) \]

5. **Cancel \(C_w\)**:
Since \(C_w\) is the same on both sides, we can cancel it out:
\[ m_1 \cdot (T_f - T_1) = m_2 \cdot (T_2 - T_f) \]

6. **Substitute known values**:
Substitute \(m_1 = 10 \, \text{kg}\), \(T_f = 30 \, \text{°C}\), \(T_1 = 12 \, \text{°C}\), and \(T_2 = 50 \, \text{°C}\):
\[ 10 \cdot (30 - 12) = m_2 \cdot (50 - 30) \]

7. **Calculate the left side**:
\[ 10 \cdot 18 = m_2 \cdot 20 \]
\[ 180 = m_2 \cdot 20 \]

8. **Solve for \(m_2\)**:
\[ m_2 = \frac{180}{20} = 9 \, \text{kg} \]

### Final Answer:
The amount of water at 50°C that should be added is **9 kg**.