To solve this problem, we need to apply the principle of energy balance, which states that the heat lost by the hot water will be equal to the heat gained by the cold water. Here is a detailed, step-by-step solution:
1. Define the known quantities:
- Temperature of hot water, [tex]\( T_{\text{hot}} = 100^\circ C \)[/tex]
- Temperature of cold water, [tex]\( T_{\text{cold}} = 15^\circ C \)[/tex]
- Required bath water temperature, [tex]\( T_{\text{final}} = 40^\circ C \)[/tex]
- Mass of hot water, [tex]\( m_{\text{hot}} = 8 \)[/tex] kg
2. Set up the principle of energy balance:
The heat lost by the hot water will be equal to the heat gained by the cold water.
3. Express the energy balance in terms of mass and temperature:
[tex]\[
m_{\text{hot}} \cdot (T_{\text{hot}} - T_{\text{final}}) = m_{\text{cold}} \cdot (T_{\text{final}} - T_{\text{cold}})
\][/tex]
Here, [tex]\( m_{\text{cold}} \)[/tex] is the unknown mass of cold water that we need to find.
4. Substitute the known values into the equation:
[tex]\[
8 \cdot (100 - 40) = m_{\text{cold}} \cdot (40 - 15)
\][/tex]
5. Simplify the temperatures:
[tex]\[
8 \cdot 60 = m_{\text{cold}} \cdot 25
\][/tex]
6. Solve for the mass of cold water:
[tex]\[
m_{\text{cold}} = \frac{8 \cdot 60}{25}
\][/tex]
7. Calculate the result:
[tex]\[
m_{\text{cold}} = \frac{480}{25} = 19.2 \text{ kg}
\][/tex]
Therefore, the mass of cold water at [tex]\(15^\circ C\)[/tex] that needs to be added to 8 kg of water at [tex]\(100^\circ C\)[/tex] to achieve the desired bath water temperature of [tex]\(40^\circ C\)[/tex] is [tex]\(19.2\)[/tex] kg.
