Question: A bio-plastic formula uses a mixture of two compounds, A and B, such that the ratio of A to B is $ \fracab $, and the effectiveness is modeled by $ E = a^2 + b^2 - ab $. If $ a = 2b $, what is the value of $ E $? - Parker Core Knowledge

Understanding the Context

The performance of this bio-plastic is defined by a mathematical function that integrates two primary components: compound A and compound B. The effectiveness, denoted by $ E $, is modeled by:

$$
E = a^2 + b^2 - ab
$$

Where:

• $ a $ represents the concentration (or effectiveness contribution) of compound A
  
• $ b $ represents the concentration of compound B

This formula suggests a quadratic relationship with interaction effects between the two compounds—capturing how synergy or conflict in composition affects the final material.

Key Insights

Applying the Given Ratio: $ a = 2b $

Rather than treating $ a $ and $ b $ as independent variables, this model assumes a fixed ratio between them. Substituting $ a = 2b $ into the effectiveness equation gives:

$$
E = (2b)^2 + b^2 - (2b)(b)
$$

First, calculate each term:

• $ (2b)^2 = 4b^2 $
  
• $ b^2 = b^2 $
  
• $ (2b)(b) = 2b^2 $

Final Thoughts

Now substitute back:

$$
E = 4b^2 + b^2 - 2b^2 = (4 + 1 - 2)b^2 = 3b^2
$$

Thus, the effectiveness $ E $ depends solely on $ b^2 $, scaled by a factor of 3. This confirms that under the $ a = 2b $ ratio, the performance of the bio-plastic is directly proportional to the square of the concentration of compound B—offering a tunable design path for material scientists.

Why This Matters in Bio-Plastic Development

By fixing the ratio between $ a $ and $ b $, researchers reduce complexity while enabling precise control over material properties. This kind of model supports sustainable innovation by:

• Minimizing trial-and-error in formulation
  
• Enhancing reproducibility and scalability
  
• Facilitating environmentally friendly performance optimization