The distance d, in mills , that an object travels at a uniform speed is directly proportional to the number of hours t it travels . If the object travels 6 miles in 2 hours , which could be the graph of the relationship between d and t ?

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to establish the relationship between distance \( d \) and time \( t \) given that the distance is directly proportional to time.

### Step-by-Step Solution:

1. **Understanding Direct Proportionality**:
The relationship between distance \( d \) and time \( t \) can be expressed as:
\[ d = kt \]
where \( k \) is the constant of proportionality (speed).

2. **Finding the Constant of Proportionality**:
We know from the problem that the object travels 6 miles in 2 hours. We can use this information to find \( k \):
\[ 6 = k \cdot 2 \]
To find \( k \), we divide both sides by 2:
\[ k = \frac{6}{2} = 3 \]

3. **Establishing the Equation**:
Now that we have \( k \), we can write the equation that relates distance and time:
\[ d = 3t \]

4. **Analyzing the Graph**:
The equation \( d = 3t \) indicates that for every hour \( t \), the distance \( d \) increases linearly. Specifically:
- When \( t = 0 \), \( d = 0 \).
- When \( t = 1 \), \( d = 3 \).
- When \( t = 2 \), \( d = 6 \).
- When \( t = 3 \), \( d = 9 \).
This means the graph will be a straight line starting from the origin (0,0) and increasing with a slope of 3.

5. **Identifying Possible Graphs**:
We need to look for a graph that starts at the origin and has a positive slope. Any graph that has negative values for time or does not start at the origin can be eliminated.

6. **Conclusion**:
After analyzing the options, the correct graph will be the one that represents a linear relationship starting from the origin with a slope of 3, which corresponds to option B.