https://file.aiquickdraw.com/mathbot/file/1726185842003ziso4c46.jpg

The image presents the following mathematical relation:

"The relation between two numbers \( a \) and \( y \) is \( a = y^2 + 10 \), and \( y \) is increased by a corresponding percentage increase in \( a \)."

### Understanding the Problem:

The equation is \( a = y^2 + 10 \). 
If \( y \) is increased, we need to determine how \( a \) will change and find the corresponding percentage change.

### Step-by-step Analysis:

1. **Differentiate the equation**:
   - We need to differentiate \( a = y^2 + 10 \) with respect to \( y \), to find how a small change in \( y \) (\( \Delta y \)) affects \( a \).
   \[
   \frac{da}{dy} = 2y
   \]

2. **Relative percentage increase**:
   - If \( y \) is increased by a small percentage, say \( \Delta y \), we need to relate this to the percentage increase in \( a \).
   - Using differentials:
     \[
     \frac{da}{a} = \frac{2y \, dy}{y^2 + 10}
     \]

This shows how the percentage change in \( a \) depends on \( y \) and the initial relation.

Would you like further details or clarifications? Here are five related questions to expand on this:

1. What is the percentage increase in \( a \) for a 10% increase in \( y \)?
2. How does the change in \( a \) compare to the change in \( y \) for small values of \( y \)?
3. Can we express the percentage change in \( a \) as a function of the percentage change in \( y \)?
4. What happens to the relationship when \( y \) becomes very large?
5. How do we calculate the total percentage increase in \( a \) after several increases in \( y \)?

**Tip**: Always differentiate the equation to find how one variable affects another when they are connected through an equation.

The relation between two numbers a and y is a = y^2 + 10. y is increased by a certain percentage. Determine the corresponding percentage increase in a.

Understand how to calculate the corresponding percentage increase in 'a' when 'y' is increased in the equation a = y^2 + 10. Learn differentiation techniques and their application to percentage changes in algebraic relations, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation