The correct curve between `log_(e) R` and `log_(e) (theta - theta_(o)`) is -

Options

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Answer

To solve the problem, we need to establish the relationship between the logarithm of the rate of cooling (log_e R) and the logarithm of the difference between the temperature of the body (theta) and the surrounding temperature (theta_o).

### Step-by-Step Solution:

1. **Understanding the Variables**:
- Let \( R \) be the rate of cooling, defined as \( R = -\frac{dT}{dt} \) (the negative sign indicates cooling).
- Let \( \theta \) be the absolute temperature of the body.
- Let \( \theta_o \) be the temperature of the surroundings.

2. **Applying Newton's Law of Cooling**:
- According to Newton's Law of Cooling, the rate of cooling can be expressed as:
\[ R = k (\theta - \theta_o) \]
where \( k \) is a constant of proportionality.

3. **Taking the Natural Logarithm**:
- We take the natural logarithm (log_e) of both sides:
\[ \log_e R = \log_e [k (\theta - \theta_o)] \]
- Using the properties of logarithms (specifically the product rule), we can expand the right side:
\[ \log_e R = \log_e k + \log_e (\theta - \theta_o) \]

4. **Rearranging the Equation**:
- This can be rearranged to:
\[ \log_e R = \log_e (\theta - \theta_o) + \log_e k \]
- This is in the form of a linear equation \( y = mx + c \), where:
- \( y = \log_e R \)
- \( x = \log_e (\theta - \theta_o) \)
- \( m = 1 \) (the slope)
- \( c = \log_e k \) (the y-intercept)

5. **Analyzing the Graph**:
- Since \( k \) is a positive constant, \( \log_e k \) is also positive.
- The slope \( m = 1 \) indicates that the graph is a straight line with a positive slope.
- The y-intercept is positive, confirming that the line will cross the y-axis above the origin.

6. **Conclusion**:
- The relationship between \( \log_e R \) and \( \log_e (\theta - \theta_o) \) is linear, with a positive slope and a positive intercept. Therefore, the correct curve is a straight line that rises from left to right.

### Final Answer:
The correct curve between \( \log_e R \) and \( \log_e (\theta - \theta_o) \) is a straight line with a positive slope and a positive y-intercept.