Which of the following graphs has a positive , decreasing slope ?

Options

A.
B.
C.
D.

Answer

To determine which graph has a positive, decreasing slope, we need to analyze the properties of the slope in relation to the angle it forms with the x-axis.

### Step-by-Step Solution:

1. **Understanding Slope**: The slope of a graph is defined as the tangent of the angle (θ) that the graph makes with the x-axis. Mathematically, slope (m) = tan(θ).

2. **Condition for Positive Slope**: For the slope to be positive, the angle θ must be between 0° and 90°. This means that as long as the graph is rising, the slope will be positive.

3. **Condition for Decreasing Slope**: A decreasing slope means that as we move along the graph, the angle θ is decreasing. This implies that the tangent of the angle (and hence the slope) is getting smaller.

4. **Analyzing the Options**:
- **Option 1**: If the slope is constant, then θ does not change, meaning the slope is not decreasing.
- **Option 2**: If the slope is increasing (θ is increasing), then the slope cannot be decreasing.
- **Option 3**: If we draw tangents at different points on this graph and observe that the angle θ is decreasing as we move along the graph, then the slope is positive and decreasing.
- **Option 4**: If the slope is negative, then it does not satisfy the condition of being positive.

5. **Conclusion**: Based on the analysis, the graph that has a positive and decreasing slope is **Option 3**.