A cup of tea cools from `65.5^@`C to `62.55^@C` in one minute is a room at `225 .^@C.` How long will the same cup of tea take to cool from `46.5^@C` to `40.5^@C` in the same room ? (Choose the nearest value in min).

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem of how long it will take for the cup of tea to cool from 46.5°C to 40.5°C in a room at 22.5°C, we can use Newton's Law of Cooling. Here are the steps to find the solution:

### Step 1: Identify the temperatures
- Initial temperature (T1) = 46.5°C
- Final temperature (T2) = 40.5°C
- Room temperature (T0) = 22.5°C

### Step 2: Use 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 temperature and the ambient temperature. The formula can be expressed as:

\[ \frac{T1 - T2}{t} = -K \left( \frac{T1 + T2}{2} - T0 \right) \]

### Step 3: Set up the equation for the first cooling scenario
From the problem, we know that the cup of tea cools from 65.5°C to 62.5°C in 1 minute. We can set up the equation:

- T1 = 65.5°C
- T2 = 62.5°C
- Room temperature (T0) = 22.5°C
- Time taken (t) = 1 minute

Substituting these values into the equation:

\[ \frac{65.5 - 62.5}{1} = -K \left( \frac{65.5 + 62.5}{2} - 22.5 \right) \]

Calculating the left side:

\[ 3 = -K \left( \frac{128}{2} - 22.5 \right) = -K (64 - 22.5) = -K (41.5) \]

### Step 4: Solve for K
Rearranging the equation gives:

\[ 3 = -K \cdot 41.5 \]

Thus,

\[ K = -\frac{3}{41.5} \]

### Step 5: Set up the equation for the second cooling scenario
Now we need to find the time taken for the cup of tea to cool from 46.5°C to 40.5°C. Using the same formula:

\[ \frac{46.5 - 40.5}{t} = -K \left( \frac{46.5 + 40.5}{2} - 22.5 \right) \]

Calculating the left side:

\[ 6 = -K \left( \frac{87}{2} - 22.5 \right) = -K (43.5 - 22.5) = -K (21) \]

### Step 6: Substitute K into the second equation
Substituting the value of K from Step 4 into the second equation:

\[ 6 = -\left(-\frac{3}{41.5}\right) \cdot 21 \]

This simplifies to:

\[ 6 = \frac{3 \cdot 21}{41.5} \]

### Step 7: Solve for t
Now we can rearrange to find t:

\[ t = \frac{6 \cdot 41.5}{3 \cdot 21} \]

Calculating this gives:

\[ t = \frac{249}{63} \approx 3.95 \text{ minutes} \]

### Step 8: Round to the nearest value
Rounding 3.95 to the nearest whole number gives approximately 4 minutes.

### Final Answer
The time taken for the cup of tea to cool from 46.5°C to 40.5°C is approximately **4 minutes**.

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