91Ó°ÊÓ

Inverse variation is an important concept in mathematics, often encountered in various fields such as physics, economics, and engineering. Inverse variation describes a relationship where the product of two variables is constant. This means that as one variable increases, the other decreases proportionally to keep their product the same. This relationship is expressed mathematically as: \[ y = \frac{k}{x} \] where \(k\) is a constant. In the given exercise, \(B\) and \(Z\) are inversely proportional to \(x^4\) and \(x^p\) respectively. Understanding inverse variation helps predict how changes in one variable affect another. When \(x\) is doubled, because \(B\) and \(Z\) are in inverse relationships with powers of \(x\), these variables respond by being reduced by a certain factor.

Mathematical relationships are fundamental to understanding numerous concepts across different domains. They describe how one quantity changes with respect to another. The relationship can be linear, quadratic, exponential, or, as in this exercise, inverse. Inverse relationships specifically denote situations where one variable increases while the other decreases.
For example:

• In part (a), \(B\) is inversely proportional to \(x^4\).
• In part (b), \(Z\) is inversely proportional to \(x^p\), where \(p\) is any positive integer.

By understanding these relationships, we can determine how changing one variable affects another. Doubling \(x\) impacts \(B\) and \(Z\) differently based on their particular inverse relationship expressions.

The constant of proportionality is a key concept that links variables in proportional relationships. In inverse relationships, this constant remains unchanged even when variables vary. It represents the fixed product of the two variables. In our examples:

• For \(B\): \( B = \frac{k}{x^4} \)
• For \(Z\): \( Z = \frac{k}{x^p} \)

Here, \(k\) is the constant of proportionality. When \(x\) is doubled, the new equations become:

• \( B_{new} = \frac{k}{(2x)^4} = \frac{k}{16x^4} \) for \(B\)
• \( Z_{new} = \frac{k}{(2x)^p} = \frac{k}{2^p \cdot x^p} \) for \(Z\)

This demonstrates that \(k\) remains constant, but \(B\) and \(Z\) are adjusted by other factors \(16\) and \(2^p\) respectively, due to the change in \(x\). Understanding this helps in predicting how the variables interdependently adjust.