If a body takes 6 minutes to cool from 80°C to 70°C, how long will it take to cool from 60°C to 50°C, if the surrounding temperature is 25°C?

Answer

To solve the problem of how long it will take for a body to cool from 60°C to 50°C, given that it takes 6 minutes to cool from 80°C to 70°C in a surrounding temperature of 25°C, we will use Newton's Law of Cooling.

### Step-by-Step Solution:

1. **Understand Newton's Law of Cooling**:
According to Newton's Law of Cooling, the rate of change of temperature of an object is proportional to the difference between its own temperature and the ambient temperature. The formula can be expressed as:
\[ \frac{dT}{dt} = -k(T - T_0) \]
where \( T \) is the temperature of the body, \( T_0 \) is the surrounding temperature, and \( k \) is a constant.

2. **Set Up the First Case**:
For the first case, where the body cools from 80°C to 70°C:
- Initial temperature \( T_1 = 80°C \)
- Final temperature \( T_2 = 70°C \)
- Surrounding temperature \( T_0 = 25°C \)
- Time taken \( t_1 = 6 \) minutes

The temperature difference for this case is:
\[ \Delta T_1 = T_1 - T_0 = 80 - 25 = 55°C \]
\[ \Delta T_2 = T_2 - T_0 = 70 - 25 = 45°C \]

3. **Calculate the Average Temperature**:
The average temperature during this cooling process can be approximated as:
\[ T_{avg1} = \frac{T_1 + T_2}{2} = \frac{80 + 70}{2} = 75°C \]

4. **Set Up the Second Case**:
For the second case, where the body cools from 60°C to 50°C:
- Initial temperature \( T_3 = 60°C \)
- Final temperature \( T_4 = 50°C \)

The temperature difference for this case is:
\[ \Delta T_3 = T_3 - T_0 = 60 - 25 = 35°C \]
\[ \Delta T_4 = T_4 - T_0 = 50 - 25 = 25°C \]

5. **Calculate the Average Temperature for the Second Case**:
The average temperature during this cooling process can be approximated as:
\[ T_{avg2} = \frac{T_3 + T_4}{2} = \frac{60 + 50}{2} = 55°C \]

6. **Set Up the Ratio of Cooling Times**:
Using the cooling law, we can set up the ratio of the time taken for the two cases:
\[ \frac{t_1}{t_2} = \frac{\Delta T_1}{\Delta T_3} \cdot \frac{T_{avg2}}{T_{avg1}} \]
Substituting the known values:
\[ \frac{6}{t_2} = \frac{10}{10} \cdot \frac{55}{75} \]
Simplifying gives:
\[ \frac{6}{t_2} = \frac{55}{75} \]

7. **Solve for \( t_2 \)**:
Rearranging gives:
\[ t_2 = 6 \cdot \frac{75}{55} \]
\[ t_2 = 6 \cdot \frac{15}{11} \approx 8.18 \text{ minutes} \]

### Final Answer:
It will take approximately **8.18 minutes** for the body to cool from 60°C to 50°C.