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Explain why the energy output is not proportional to the number of days.

To determine whether the energy output (\( E \)) is proportional to the number of days (\( d \)), we need to check if there is a constant ratio between \( E \) and \( d \) across all data points in the table.

Let's examine the ratios of \( \frac{E}{d} \) for each pair in the table:

1. For \( E = 1200 \) and \( d = 2.4 \):
   \[
   \frac{E}{d} = \frac{1200}{2.4} = 500
   \]

2. For \( E = 1800 \) and \( d = 3.6 \):
   \[
   \frac{E}{d} = \frac{1800}{3.6} = 500
   \]

3. For \( E = 4000 \) and \( d = 8 \):
   \[
   \frac{E}{d} = \frac{4000}{8} = 500
   \]

4. For \( E = 10000 \) and \( d = 20 \):
   \[
   \frac{E}{d} = \frac{10000}{20} = 500
   \]

Since each calculation gives us the same ratio, \( \frac{E}{d} = 500 \), we can see that \( E \) is indeed proportional to \( d \). This means there *is* a constant relationship, which can be represented by the equation:

\[
E = 500d
\]

So, contrary to what was initially asked, the energy output actually *is* proportional to the number of days, with the proportional constant of \( 500 \). 

If you have further questions about this, or need additional details, please let me know!

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### Additional Questions
1. What would the energy output be after 10 days using this relationship?
2. How would you describe the constant of proportionality in this scenario?
3. If the number of days was doubled, what would happen to the energy output?
4. Can you rearrange the equation \( E = 500d \) to express \( d \) in terms of \( E \)?
5. How could you graph this proportional relationship between \( E \) and \( d \)?

### Tip
When checking for proportionality, always ensure the ratio between two quantities remains constant across all data points.

Explore the proportional relationship between energy output (E) and the number of days (d) based on data from a coal power plant. Understand the equation E = 500d, which shows a constant ratio of 500 MW per day. Suitable for students in Grades 7-9.

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