The height of the steam burst of certain geyser varies with the length of time since the previous steam burst. The longer the time since the last burst, the greater the height of the steam burst. If t is the time in hours since the previous steam burst and H is the height in meters of the steam burst, which of the following could express the relationship of t and H?

Options

A. `H(t)=(1)/(2)(t-7)`
B. `H(t)=(2)/(t-7)`
C. `H(t)=2-(t-7)`
D. `H(t)=7-2t`

Answer

To solve the problem, we need to establish the relationship between the time since the previous steam burst (t) and the height of the steam burst (H). The problem states that the longer the time since the last burst, the greater the height of the steam burst. This indicates a direct relationship between t and H.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
- The problem states that as the time (t) increases, the height (H) also increases. This indicates a direct proportionality.
- Mathematically, we can express this relationship as:
\[ H \propto t \]
- This means that H is directly proportional to t.

2. **Expressing the Proportionality**:
- We can express the direct proportionality with an equation by introducing a constant of proportionality (k):
\[ H = k \cdot t \]
- Here, k is a positive constant since both H and t are positive quantities.

3. **Analyzing the Options**:
- We need to evaluate the given options to determine which one expresses the relationship correctly.
- Since we established that H increases with t, we should look for an option that reflects this direct relationship.

4. **Evaluating Each Option**:
- **Option A**: If it shows a positive relationship (like \(H = kt\)), it is a candidate.
- **Option B**: If it shows an inverse relationship (like \(H = \frac{2}{t - 7}\)), it is not correct.
- **Option C**: If it has a negative coefficient for t, it is not correct.
- **Option D**: Similar to option C, if it has a negative coefficient for t, it is not correct.

5. **Conclusion**:
- After evaluating the options, the only option that maintains a positive relationship between t and H is **Option A**. Therefore, it is the correct expression of the relationship.

### Final Answer:
The correct expression for the relationship between t and H is given by **Option A**.