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### Transcribed Question
7. Given that \( a = 36 \) and \( b = 49 \), if \( x \) is directly proportional to the square root of \( y \):
  (i) \( x \) when \( y = 36 \)
  (ii) \( y \) when \( x = 81 \)

### Solution

Given \( x \) is directly proportional to the square root of \( y \), we can write the relationship as:
\[ x = k \sqrt{y} \]
where \( k \) is the constant of proportionality.

1. **Find \( k \) using the given values \( a = 36 \) and \( b = 49 \):**
   \[ a = k \sqrt{b} \]
   \[ 36 = k \sqrt{49} \]
   \[ 36 = k \cdot 7 \]
   \[ k = \frac{36}{7} \]

2. **Part (i): Find \( x \) when \( y = 36 \):**
   \[ x = k \sqrt{36} \]
   \[ x = \frac{36}{7} \cdot 6 \]
   \[ x = \frac{216}{7} \]
   \[ x \approx 30.86 \]

3. **Part (ii): Find \( y \) when \( x = 81 \):**
   \[ 81 = k \sqrt{y} \]
   \[ 81 = \frac{36}{7} \sqrt{y} \]
   \[ 81 = \frac{36 \sqrt{y}}{7} \]
   \[ 81 \cdot 7 = 36 \sqrt{y} \]
   \[ 567 = 36 \sqrt{y} \]
   \[ \sqrt{y} = \frac{567}{36} \]
   \[ \sqrt{y} = 15.75 \]
   \[ y = 15.75^2 \]
   \[ y = 248.0625 \]

So the solutions are:
1. \( x \approx 30.86 \) when \( y = 36 \)
2. \( y \approx 248.06 \) when \( x = 81 \)

Would you like any further details or explanations on this problem?

### Follow-up Questions:
1. What is the proportional relationship between two variables?
2. How do you determine the constant of proportionality?
3. Can you explain the concept of direct proportionality with an example?
4. How would the relationship change if \( x \) was inversely proportional to \( \sqrt{y} \)?
5. What are some real-life examples of direct proportionality?
6. How does the square root function affect the relationship between \( x \) and \( y \)?
7. How can you verify your solutions for \( x \) and \( y \) are correct?
8. What methods can be used to solve proportional relationships involving other functions (e.g., cube roots)?

#### Tip:
When dealing with proportional relationships, always identify the constant of proportionality first, as it simplifies the process of finding unknown values.

Explore how to solve a direct proportionality problem where x is directly proportional to the square root of y. Given constants a = 36 and b = 49, find x when y = 36 and y when x = 81. Detailed step-by-step solution provided.

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