To find the ratio of the masses of hot and cold water needed to achieve a desired temperature, we can use the principle of mixing. The equation for the final temperature $T_f$ when mixing two masses of water at different temperatures is:

$m_c \cdot T_c + m_h \cdot T_h = (m_c + m_h) \cdot T_f$

Where:

• $m_c$ = mass of cold water
• $T_c$ = temperature of cold water (26 °C)
• $m_h$ = mass of hot water
• $T_h$ = temperature of hot water (82 °C)
• $T_f$ = desired temperature (40 °C)

Rearranging gives:

$m_c \cdot T_c + m_h \cdot T_h = m_c \cdot T_f + m_h \cdot T_f$

This can be rewritten as:

$m_h \cdot T_h = m_c \cdot T_f + m_h \cdot T_f - m_c \cdot T_c$

Grouping terms leads to:

$m_h \cdot (T_h - T_f) = m_c \cdot (T_f - T_c)$

Now, we can express the ratio of masses:

$\frac{m_h}{m_c} = \frac{T_f - T_c}{T_h - T_f}$

Substituting in the values:

• $T_f = 40$
• $T_c = 26$
• $T_h = 82$

This gives:

$\frac{m_h}{m_c} = \frac{40 - 26}{82 - 40} = \frac{14}{42} = \frac{1}{3}$

Thus, the ratio of the mass of hot water to that of cold water is:

$\frac{m_h}{m_c} = \frac{1}{3}$

Summary

The required ratio of the mass of hot water to that of cold water is $1:3$.

Would you like more details or have any questions? Here are some related questions you might consider:

1. How does the specific heat capacity of water affect this calculation?
2. What would happen if the desired temperature was higher than 82 °C?
3. How would the ratio change if the cold water temperature were lower?
4. Can you calculate the total volume of water needed for the bath?
5. What are the practical implications of mixing water at these temperatures?

Tip: Always ensure that the temperatures used in calculations are within the limits of the materials involved to avoid any practical issues, like boiling or freezing.