https://file.aiquickdraw.com/mathbot/file/17272817511723gh243ug.jpg

The problem in the image states that \( b \) is directly proportional to the square root of \( a \), and asks you to complete the table.

We can express the relationship mathematically as:
\[
b = k \sqrt{a}
\]
where \( k \) is the constant of proportionality.

### Step 1: Find \( k \)
Using the values \( a = 1 \) and \( b = 9 \) from the table, we substitute these into the equation:
\[
9 = k \sqrt{1}
\]
Since \( \sqrt{1} = 1 \), we find:
\[
k = 9
\]

### Step 2: Use \( k \) to complete the table
Now that we know \( k = 9 \), we can use this to find the missing values for \( b \) and the corresponding \( a \).

- When \( a = 36 \):
\[
b = 9 \times \sqrt{36} = 9 \times 6 = 54
\]

So the missing value for \( b \) when \( a = 36 \) is **54**.

Thus, the completed table looks like this:

| \( a \) | 1  | 36 |  
| \( b \) | 9  | 54 |

Would you like any further clarification on this solution? Here are five related questions you might find useful:

1. What if \( b \) were inversely proportional to the square root of \( a \)?
2. How would the table change if \( b \) were proportional to \( a^2 \) instead of \( \sqrt{a} \)?
3. Can you graph the relationship between \( a \) and \( b \) for this proportional relationship?
4. What are other examples of directly proportional relationships in mathematics?
5. How do you determine the constant of proportionality in any relationship?

**Tip**: Always ensure to substitute known values into the proportionality equation to find the constant before using it for other calculations.

b is directly proportional to the square root of a. Complete the table.

Learn how to complete the table where b is directly proportional to the square root of a. This problem explores direct proportionality and square roots, and provides a detailed solution, including finding the constant of proportionality. Suitable for students in Grades 8-10.

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