A hot liquid is kept in a big room. According to Newton's law of cooling rate of cooling liquid (represented as y) is plotted against its temperature `T`. Which of the following curves may represent the plot?

Options

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B.
C.
D.

Answer

To solve the problem, we will analyze the situation using Newton's Law of Cooling, which states that the rate of cooling of an object is proportional to the difference in temperature between the object and its surroundings.

### Step-by-Step Solution:

1. **Understanding Newton's Law of Cooling**:
According to Newton's Law of Cooling, the rate of change of temperature of the hot liquid (denoted as \( y \)) is given by:
\[ \frac{dT}{dt} = -k(T - T_s) \]
where:
- \( T \) is the temperature of the liquid,
- \( T_s \) is the surrounding temperature (constant),
- \( k \) is a positive constant.

2. **Rearranging the Equation**:
We can rearrange the equation to express the rate of cooling \( y \) in terms of the temperature \( T \):
\[ y = -k(T - T_s) \]
This can be rewritten as:
\[ y = -kT + kT_s \]
This equation is in the form of a linear equation \( y = mx + c \), where:
- \( m = -k \) (the slope),
- \( c = kT_s \) (the y-intercept).

3. **Analyzing the Graph**:
- The slope \( m \) is negative, indicating that as the temperature \( T \) increases, the rate of cooling \( y \) decreases.
- The y-intercept \( c \) is positive since \( kT_s \) is positive (assuming \( T_s \) is a positive temperature).

4. **Identifying the Correct Curve**:
- Since the relationship is linear with a negative slope, the graph will be a straight line that slopes downwards from left to right.
- The line will intersect the y-axis at a positive value (the y-intercept) and will extend downwards as \( T \) increases.

5. **Conclusion**:
Based on the analysis, the correct curve that represents the plot of the rate of cooling \( y \) against the temperature \( T \) is a straight line with a negative slope that intersects the y-axis at a positive value.