A body cools in 7 min from `60^@C` to `40^@C` What will be its temperature after the next 7 min? The temperature of surroundings is `10^@C`.

Options

A. `28^@C`
B. `25^@C`
C. `30^@C`
D. `22^@C`

Explanation

To solve the problem of a body cooling from 60°C to 40°C in 7 minutes, and to find its temperature after the next 7 minutes, we will use Newton's Law of Cooling. Here’s a step-by-step solution:

### Step 1: Understand Newton's Law of Cooling
Newton's Law of Cooling states that the rate of change of temperature of an object is directly proportional to the difference between its temperature and the ambient (surrounding) temperature. Mathematically, it can be expressed as:
\[ \frac{dT}{dt} \propto (T - T_s) \]
where \( T \) is the temperature of the body, and \( T_s \) is the temperature of the surroundings.

### Step 2: Set Up the Initial Conditions
We know that the body cools from 60°C to 40°C in 7 minutes. The surrounding temperature \( T_s \) is given as 10°C.

### Step 3: Calculate the Average Temperature During the First Interval
The average temperature during the first cooling period can be calculated as:
\[ T_{avg1} = \frac{T_1 + T_2}{2} = \frac{60 + 40}{2} = 50°C \]

### Step 4: Write the Cooling Equation for the First Interval
Using Newton's Law of Cooling, we can express the change in temperature over the first 7 minutes:
\[ \frac{T_2 - T_1}{\Delta t} = -k(T_{avg1} - T_s) \]
Substituting the values:
\[ \frac{40 - 60}{7} = -k(50 - 10) \]
This simplifies to:
\[ \frac{-20}{7} = -k(40) \]
Thus:
\[ k = \frac{20}{7 \times 40} = \frac{1}{14} \]

### Step 5: Set Up the Conditions for the Second Interval
Now, we need to find the temperature after the next 7 minutes. Let’s denote the temperature after the first 7 minutes as \( T_0 \) (which is 40°C). The average temperature for the next interval will be:
\[ T_{avg2} = \frac{T_0 + 10}{2} = \frac{40 + 10}{2} = 25°C \]

### Step 6: Write the Cooling Equation for the Second Interval
Using the same form of the equation for the second interval:
\[ \frac{T_3 - T_0}{\Delta t} = -k(T_{avg2} - T_s) \]
Substituting the known values:
\[ \frac{T_3 - 40}{7} = -\frac{1}{14}(25 - 10) \]
This simplifies to:
\[ \frac{T_3 - 40}{7} = -\frac{1}{14}(15) \]
\[ \frac{T_3 - 40}{7} = -\frac{15}{14} \]
Multiplying both sides by 7:
\[ T_3 - 40 = -\frac{15 \times 7}{14} \]
\[ T_3 - 40 = -\frac{105}{14} \]
\[ T_3 - 40 = -7.5 \]
Thus:
\[ T_3 = 40 - 7.5 = 32.5°C \]

### Step 7: Conclusion
The temperature of the body after the next 7 minutes will be approximately **32.5°C**.

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