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The given relation between the two variables \( q \) and \( y \) is:

\[
q = y^2 + 10
\]

The problem states that \( y \) is increased by the corresponding percentage increase in \( q \).

To solve this, let's break down what happens when \( y \) increases and how the percentage change in \( q \) corresponds to the percentage change in \( y \).

1. **Differentiate the relation to find the rate of change of \( q \) with respect to \( y \):**

   \[
   \frac{dq}{dy} = 2y
   \]

2. **Percentage increase in \( q \):**
   The percentage change in \( q \) is given by:

   \[
   \text{Percentage increase in } q = \frac{\Delta q}{q} \times 100
   \]

   where \( \Delta q \) is the change in \( q \) and \( q = y^2 + 10 \).

3. **Percentage increase in \( y \):**
   The percentage change in \( y \) can similarly be written as:

   \[
   \text{Percentage increase in } y = \frac{\Delta y}{y} \times 100
   \]

4. **Relating the changes:**
   Since \( y \) increases by the same percentage as \( q \), you can equate these expressions to find the relationship between \( \Delta q \) and \( \Delta y \).

Would you like me to continue the solution by applying these steps or clarify a specific part? Here are five related questions:

1. What is the importance of differentiating in this context?
2. How can we express the change in \( y \) based on the given percentage change in \( q \)?
3. If the percentage change in \( q \) is known, how do we calculate the new value of \( y \)?
4. Can this method be applied to other forms of relationships between two variables?
5. How would the result change if the relationship between \( q \) and \( y \) were more complex?

**Tip:** When working with percentage changes in related variables, always consider using derivatives to connect their rates of change.

The relation between two numbers q and y is q = y^2 + 10, y is increased by corresponding percentage increase in q.

Explore how to relate percentage increases in q and y, where q = y^2 + 10. Understand the use of differentiation and percentage change formulas to solve this problem. Suitable for Grades 9-12.

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