A body in a room cools from `85^(@)`C to `80^(@)` C in 5 minutes. Calculate the time taken to cool from `80^(@)` C to `75^(@)` C if the surrounding temperature is `30^(@)`C.

Answer

To solve the problem, we will use Newton's Law of Cooling, which states that the rate of change of temperature of an object is proportional to the difference between its temperature and the ambient temperature.

### Step-by-Step Solution

1. **Identify the Given Information**:
- Initial temperature (T1) = 85°C
- Final temperature (T2) = 80°C
- Time taken to cool from T1 to T2 (Δt1) = 5 minutes
- Surrounding temperature (T0) = 30°C

2. **Calculate the Change in Temperature for the First Interval**:
- Change in temperature (ΔT1) = T1 - T2 = 85°C - 80°C = 5°C

3. **Calculate the Average Temperature for the First Interval**:
- Average temperature (T_avg1) = (T1 + T2) / 2 = (85°C + 80°C) / 2 = 82.5°C

4. **Set Up the Equation Using Newton's Law of Cooling for the First Interval**:
- According to Newton's Law of Cooling:
\[ \frac{ΔT1}{Δt1} = k (T_{avg1} - T0) \]
- Plugging in the values:
\[ \frac{5}{5} = k (82.5 - 30) \]
- Simplifying gives:
\[ 1 = k \cdot 52.5 \quad \Rightarrow \quad k = \frac{1}{52.5} \]

5. **Calculate the Change in Temperature for the Second Interval**:
- Now, we need to calculate the time taken to cool from 80°C to 75°C.
- Change in temperature (ΔT2) = T2 - T3 = 80°C - 75°C = 5°C

6. **Calculate the Average Temperature for the Second Interval**:
- Average temperature (T_avg2) = (T2 + T3) / 2 = (80°C + 75°C) / 2 = 77.5°C

7. **Set Up the Equation Using Newton's Law of Cooling for the Second Interval**:
- According to Newton's Law of Cooling:
\[ \frac{ΔT2}{Δt2} = k (T_{avg2} - T0) \]
- Plugging in the values:
\[ \frac{5}{Δt2} = k (77.5 - 30) \]
- Substituting k from the previous calculation:
\[ \frac{5}{Δt2} = \frac{1}{52.5} \cdot 47.5 \]

8. **Solve for Δt2**:
- Rearranging gives:
\[ Δt2 = \frac{5 \cdot 52.5}{47.5} \]
- Calculating the right side:
\[ Δt2 = \frac{262.5}{47.5} \approx 5.52 \text{ minutes} \]

### Final Answer:
The time taken to cool from 80°C to 75°C is approximately **5.52 minutes**.